SILPAKORN UNIVERSITY Science and Technology Journal SUSTJ
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Chapter in a book Langer, T. and Neupert, W. (1994) Chaperoning mitochondrial biogenesis. In The Biology of Heat Shock Proteins and Molecular Chaperones (Morimoto, R. I., Tissieres, A. and Georgopoulos, C., eds.), pp. 53-83. Cold Spring Harbor Laboratory Press, Plainview, New York. Article in a journal Hammerschlag, F. A., Bauchan, G., and Scorza, R. (1985) Regeneration of peach plants from callus derived from immature embryos. Journal of Natural Products 70(3): 248-251. Hammerschlag, F. A., Bauchan, G., and Scorza, R. Regeneration of peach plants from callus derived from immature embryos. Journal of Natural Products (in press). Article on the web Lee, K. (1999) Appraising adaptaive management. Conservation Ecology 3(2). [Online URL:www. consecolo.org/Journal/vol3/iss2/index.html] accessed on April 13, 2001. Proceedings MacKinnon, R. (2003) Modelling water uptake and soluble solids losses by puffed breakfast cereal immersed in water or milk. In Proceedings of the Seventh International Congress on Engineering and Food, Brighton, UK. Patent Yoshikawa, T. and Kawai, M. (2006) Security robot. U.S. Patent No. 2006079998. Tables and Figures Each Table and Figure must be on a separate page of the manuscript. Tables: Number the tables according to their sequence in the text. The text should include references to all tables. Vertical lines should not be used to separate columns. Leave some extra space instead. Figures: Figures should be of high quality (not less than 300 dpi JPEG or TIFF format), in black and white only, with the same size as the author would like them to appear in press. Choose the size of symbols and lettering so that the figures can be reduced to fit on a page or in a column. Submission of Manuscripts Authors should verify, on the submission form, that the submitted manuscript has not been published or is being considered for publication elsewhere. All information contained in an article is full responsibility of the authors, including the accuracy of the data and resulting conclusion. Authors are requested to send the manuscript on a CD labeled with the authors’ names and file names. The files should be prepared using MS Word only. Three copies of manuscript must be supplied. The editorial office will acknowledge receipt of the manuscript within 2 weeks of submission. The ‘accepted date’ that appears in the published article will be the date when the managing editor receives the fully revised version of the manuscript. The manuscript may be returned to authors for revision. Authors will be given 2 weeks after receipt of the reviewers’ comments to revise the article.
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Silpakorn University Science and Technology Journal
Contents
Volume 5 Number 1 (January - June) 2011
Research Articles
A Comparison of Image Analysis Software for Quantitative TLC of Ceftriaxone Sodium…….………
Panadda Phattanawasin, Uthai Sotanaphun, Lawan Sriphong,
Inthira Kanchanaphibool and Nusara Piyapolrungroj
Differentiation of a Hyperthermophilic Archaeon Pyrococcus sp. strain Pikanate 5017,
by Arbitrarily Primed PCR…………………………………………………………………………………………...………………………………………..........................
14
Patlada Pasomsup, Juan Miguel González, Maria Carmen Portillo, Veeranun Pongsapukdee and Wirojne Kanoksilapatham
All Congruence Modular Symmetric and Near-Symmetric Algebras…………………...……………………..........................
7
24
Chawewan Ratanaprasert and Supharat Thiranantanakorn
Studies on Electrochromism of Chemically Deposited Nickel Oxide Thin Films …………………….…………..
Julijana Velevska, Margareta Pecovska-Gjorgjevich,
Metodija Najdoski and Nace Stojanov
SUSTJ is now available on the following databases: Chemical Abstract Service (CAS), International Information System for the Agricultural Sciences and Technology (AGRIS), AGRICultural Online Access (AGRICOLA) Food Science and Technology Abstracts (FSTA), Directory of Open Access Journals (DOAJ), Google Scholar, Thai Journal Citation Index Centre (TCI Centre).
34
Research Article A Comparison of Image Analysis Software for Quantitative TLC of Ceftriaxone Sodium Panadda Phattanawasin,1* Uthai Sotanaphun,2 Lawan Sriphong,1 Inthira Kanchanaphibool3 and Nusara Piyapolrungroj4 1
Department of Pharmaceutical Chemistry, 2Department of Pharmacognosy, 3Department of Pharmacy, 4 Department of Biopharmacy, Faculty of Pharmacy, Silpakorn University, Nakhon Pathom, Thailand * Corresponding author. E-mail address: ypanadda@su.ac.th Received September 7, 2010; Accepted February 15, 2011
Abstract Three image analysis software, Photoshop, Sorbfil TLC Videodensitometer software and Scion Image, were used for quantitative evaluation of TLC images of ceftriaxone sodium (CFX). Regression plots and detection sensitivity of quantification from each TLC-image analysis were determined and compared to TLC-densitometry. TLC-image analysis method using Scion Image and Sorbfil TLC Videodensitometer software and TLC-densitometry showed the polynomial regression data with good relationship (R2 > 0.99) over the concentration range of 1-8 mg/spot whereas the analysis of TLC images with Photoshop showed good polynomial regression plots (R2 > 0.99) at higher concentration range, 4-10 mg/spot. For detection sensitivity, LOD and LOQ determined from TLC-image analysis using Sorbfil TLC Videodensitometer were comparable to the values from TLC-densitometry. Scion Image and Photoshop was found to be less sensitive. The use of Sorbfil TLC Videodensitometer software for TLC image analysis could be further applied for rapid determination of CFX in bulk drug and dosage forms. Key Words: TLC; Image analysis software; Ceftriaxone sodium Introduction
reversed-phase liquid chromatography, fluorimetric
Ceftriaxone sodium (CFX) is a semisynthetic,
and capillary electrophoresis methods (Gรกspรกr et
third-generation cephalosporin antibiotic which
al., 2002; Bebawy et al., 2003; El-Shaboury et al.,
has been widely used for treatment of severe
2007) have been used.
bacterial infections and bacterial meningitis. For
Until recently, the use of TLC-image analysis
determination of CFX in bulk drugs and dosage forms,
has been applied for content determination of
several chromatographic and spectrophotometric
several compounds (Hung et al., 2001; Mustoe
methods including densitometric-TLC and HPTLC
and McCrossen, 2001; Johnsson et al., 2007;
(Dhanesar, 1998; Eric-Jovanovic et al., 1998;
Sotanaphun et al., 2009). The major advantages of
Nabi et al., 2004; Mohamed et al., 2008), HPLC
TLC are due to its simplicity, a small quantity of
(Abdel-Hamid, 1998; Xu and Trissel, 2003), ion-pair
solvents used, minimum sample preparation and
Silpakorn U Science & Tech J 5 (1) : 7-13, 2011
Silpakorn U Science & Tech J Vol.5(1), 2011
A Comparison of Image Analysis Software
high sample throughput. With a combination of
tripod to a distance of 37 cm above the image. An
simple computer technology and image analysis
auto mode, ISO setting at 200 and Macro (close-up)
software for evaluation of TLC chromatogram, the
function was used and the flash was off. The shutter
quantitative TLC method based on image analysis
speed, aperture settings and zoom in were at 1/8, 5.8
is more convenient and less expensive than other
and 3.8x, respectively. The image size captured by a
chromatographic methods.
digital camera was set to superfine (S) and large (L)
Commercial and free web-based image
and saved in a form of a joint photographic experts
software for TLC-image analysis are available in
group (JPEG) file. Then, a JPEG image was resized
which performances are based on sensitivity of spot
and cropped according to the plate dimension at 10
detection, background compensation algorithms,
cm x 8 cm and saved at a resolution of 60 pixels/
intensity resolution, precision and accuracy of
cm for TLC-image analysis.
image analysis. From our attempt to develop a
TLC-Image Analysis Method
simple and rapid TLC–image analysis method for
Quantification of CFX in the TLC image
determination of CFX in bulk and pharmaceutical
was carried out by Photoshop, Sorbfil TLC
dosage forms, chromatographic separation of CFX
Videodensitometer software and Scion Image
was achieved on RP-18 F254S TLC plates. The use of
software.
three image analysis software including Photoshop,
The analysis of TLC images by Photoshop was
Sorbfil TLC Videodensitometer and Scion Image for
performed by adjusting the image at threshold value
evaluation of the TLC plates were compared with
of 130. The image was converted to black and white.
TLC-densitometric method in terms of regression
Each band was selected by a magic wand tool and
plots and detection sensitivity of quantification and
the area was determined in terms of pixel numbers.
described herein.
Pixel numbers in the selected area were measured by a histogram command in Photoshop software.
Material and Methods
For image analysis by Sorbfil TLC
Chromatographic Conditions
Videodensitometer software (Sorbpolymer,
TLC analysis was performed on TLC silica
Krasnodar, Russia), Background Approximation
gel 60 RP-18 F254S aluminium plates (10 cm x 8 cm
tool was used to provide the uniform illumination
with 185 mm thickness, Merck, Germany). Two mL
of the background and arrangement of track lines
of CFX solution were spotted manually onto a TLC
was made by using Regular Tracks command. The
plate by using a 2 mL capillary tube (Drummond
evaluation of the chosen track was processed by
Scientific Company, USA). A distance between each
using Process Track command. In order to determine
spot was 0.8-1.0 cm. The plate was then developed
the background intensity of the plate, for each line
to a distance of 6.5 cm in a TLC chamber previously
method at the values of width and height at 3 and
saturated with 15% w/v ammonium acetate buffer
30, respectively, was used. A chromatogram was
(pH 6.2)-methanol-acetonitrile (12:0.5:0.25, v/v/v)
constructed on the deviation of track intensity from
for 20 min. After air-drying, an image of the
background intensity. Rf, peak area and peak height
developed plate under a UV lamp (CAMAG UV-
of CFX were determined and the results of the track
cabinet II, Switzerland) at 254 nm was taken by
evaluation were given in the Track Evaluation
a digital camera (Ixus 860, Cannon) setting on a
Results box.
8
P. Phattanawasin et.al
Silpakorn U Science & Tech J Vol.5(1), 2011
Results and Discussion
The greyscale image of TLC chromatogram was used for analysis by Scion Image program
Reversed phase TLC (RP-18 F 254s TLC)
version Alpha 4.0.3.2 (Scion Corporation, Maryland,
plates and the mobile phase consisting of 15%
USA). A profile plot along the chromatogram was
w/v ammonium acetate buffer (pH 6.2)-methanol-
generated using the macro Gelplot2. The peak
acetonitrile (12:0.5:0.25, v/v/v) was employed to
corresponding to CFX was selected by the wand
analyze CFX, showing a well-defined spot of CFX
tool for measuring the area under the curve (AUC)
at an Rf value of 0.58. The peak area and the peak
TLC-Densitometry
height of CFX were determined from TLC-image
For TLC-densitometric analysis, the developed
analysis and TLC-densitometric method and used for
TLC plate under the chromatographic conditions
construction of regression plots. The study showed
as above was scanned by using a CAMAG TLC
that the determination of peak area and peak height
scanner II with CAMAG CATS 3.1 software in the
was available by Sorbfil TLC Videodensitometer
absorbance mode at 302 nm. The slit dimension
and TLC-densitometer whereas Photoshop and
was 5.0 mm Ă— 5.0 mm and the scanning speed was
Scion image could measure only peak area. For
4 mm/s.
peaks without full separation at peak base, peak
Regression plots
height could be more useful for determination of
Regression plots were performed by spotting
the content of substance in those peaks.
two mL of working standard solutions of CFX (0.25-
For the analysis of CFX by different TLC
5.0 mg/mL) prepared from a stock solution of CFX
methods, the results showed that the peak area of the
(5.0 mg/mL) in distilled water on a TLC plate to give
spot at concentration being less than 4 mg/spot was
the concentrations of 0.5, 1, 2, 4, 6, 8 and 10 mg/spot
unable to be detected by TLC-image analysis using
of CFX. The plate was developed according to the
Photoshop whereas both TLC-densitometric and
above TLC-image analysis and TLC-densitometric
TLC-image analysis methods using Scion Image
methods. Plots between peak height or peak area
and Sorbfil TLC Videodensitometer software could
and drug concentration in mg/spot were constructed
detect the peak corresponding to the concentration
and treated by least square regression analysis using
at 0.5 mg/spot (Figure 1). However, poor precision
Excel.
of detection was observed at this concentration.
Detection Sensitivity
Therefore, the regression data of the plots between
The linear regression data constructed from
peak height or peak area and concentrations over the range of 1-10 mg/spot were examined.
three concentrations at the lower range were used to determine limit of detection (LOD) and limit of
The data from the analysis of TLC images with
quantitation (LOQ) values (Ansari et al., 2005). The
Photoshop showed good polynomial regression plots
LOD and LOQ were determined from the formulae
(R2 > 0.99) over the concentration range of 4-10 mg/
3 SD/S and 10 SD/S where SD was the standard
spot whereas the data from both TLC-densitometric
deviation (SD) of the intercept and S corresponded
and TLC-image analysis methods using Scion
to the slope.
Image and Sorbfil TLC Videodensitometer software
9
Silpakorn U Science & Tech J Vol.5(1), 2011
A Comparison of Image Analysis Software
The image was converted to black and white and pixel numbers in the selected area were measured by a histogram command in Photoshop software. Track
1 2 3 - 4 5 6 0.5 10 1.0 2.0 1.0 4.0
7 6.0
8 8.0
mg/spot
a) before evaluation
Track
1 2 3 - 4 5 6 0.5 10 1.0 2.0 1.0 4.0
7 6.0
8 8.0 mg/spot
b) Photoshop A chromatogram was constructed for each track and the peak area was selected by the wand tool and measured.
A chromatogram was constructed from each track evaluation and the peak area and height were determined by the software. track 2 (10 mg/spot) track 8 (8 mg/spot)
track 1 (0.5 mg/spot)
track 3 (1.0 mg/spot)
track 7 (6 mg/spot) track 6 (4 mg/spot)
track 4 (2.0 mg/spot)
track 4 (2 mg/spot) track 6 (4.0 mg/spot)
track 3 (1 mg/spot) track 5 (1 mg/spot) track 1 (0.5 mg/spot)
track 7 (6.0 mg/spot)
track 8 (8.0 mg/spot)
c) Sorbfil TLC Videodensitometer software
d) Scion Image
Figure 1 TLC image a) before evaluation, and evaluation by b) Photoshop, c) Sorbfil TLC Videodensitometer software, and d) Scion Image
10
P. Phattanawasin et.al
Silpakorn U Science & Tech J Vol.5(1), 2011
Table 1 Polynomial regression data determined using TLC-densitometric and TLC-image analysis methods Analytical
Concentration
Regression
a*
b*
c*
R2
technique
m (mg/spot)
between
TLC-Densitometry
1-8
A and C
-56.072
1171.3
1435.7
0.9938
H and C
-0.7195
13.524
5.9907
0.9983
A and C
1.1625
602.03
2624.9
0.9906
H and C
-0.0875
7.475
17.86
0.9853
4-10
A and C
117.31
1513.2
-1013.4
0.9979
1-8
A and C
-2872.3
51644
-22073
0.9980
H and C
-44.705
883.99
-254.09
0.9989
A and C
58.964
23891
31358
0.9758
H and C
-5.2679
507.26
482.6
0.9880
A and C
-18.767
305.84
-101.98
0.9967
2-10 Â TLC-Image analysis using Photoshop TLC-Image analysis using Sorbfil 2-10 TLC-Image analysis *
1-8
using Scion Image 2-10 A and C 0.7321 119.66 261.8 2 polynomial regression: y = ax + bx + c; A = Area, H = Height, C = Concentration
0.9654
Table 2 Detection limits
showed the polynomial regression data with good relationship (R > 0.99) over the concentration range 2
of 1-8 mg/spot (Table 1). At higher concentration range, 2-10 mg/spot, poorer correlation coefficients were obtained.
Analytical
Regression
LOD
LOQ
technique
between
ng/spot
ng/spot
TLC-Densitometry
A and C
169
564
H and C
103
343
In order to assess and compare the sensitivity of detection limits obtained from TLC-densitometric
Image analysis
A and C
170
568
and TLC-image analysis methods using Scion
using Sorbfil
H and C
131
439
Image and Sorbfil TLC Videodensitometer, the
TLC-Image analysis
A and C
419
1397
linear regression data constructed from the lower
using Scion
concentration range (1, 2 and 4 mg/spot) were used
A = Area, H = Height, C = Concentration
to determine LOD and LOQ values (Ansari et al., 2005) (Table 2). The data showed that LOD and LOQ determined from TLC-densitometric method were
Better detection sensitivity was observed
comparable to the values from TLC-image analysis
when TLC-image analysis was performed by
method using Sorbfil TLC Videodensitometer. TLC-
Sorbfil TLC Videodensitometer. This might be due
image analysis using Scion Image was found to be
to the background approximation tool available in
less sensitive.
the software which could be used to correct the
11
Silpakorn U Science & Tech J Vol.5(1), 2011
A Comparison of Image Analysis Software
whole image brightness, thus, providing uniform
Ansari, M. J., Ahmad, S., Kohli, K., Ali, J., and Khar,
illumination of the plate before evaluation. Even
R. K. (2005) Stability-indicating HPTLC
though TLC-image analysis using Scion Image and
determination of curcumin in bulk drug and
Photoshop was found to be less sensitive, Scion
pharmaceutical formulations. Journal of
image had an advantage of being public software and
Pharmaceutical and Biomedical Analysis
free downloaded whereas Photoshop was common
39: 132-138.
commercial software and might be suitable for
Bebawy, L. L., Kelani, K. E., and Fattah, L. A.
analysis of a TLC image with uniform illumination
(2003) Fluorimetric determination of some
and spots having distinct color from a background.
antibiotics in raw material and dosage forms through ternary complex formation with
Conclusions
terbium (Tb3+). Journal of Pharmaceutical and Biomedical Analysis 32: 1219-1225.
Three image analysis software including Photoshop, Sorbfil TLC Videodensitometer software
Dhanesar, S. C. (1998) Quantitation of antibiotics by
and Scion Image software were used for quantitative
densitometry on a hydrocarbon-impregnated
evaluation of CFX from TLC images and compared
silica gel HPTLC plate. Part II: Quantitation
to TLC densitometric method. Regression data
and evaluation of ceftriaxone. Journal of
and detection sensitivity of quantification of CFX
Planar Chromatography 11: 258-262.
assessed from TLC-image analysis by Sorbfil
El-Shaboury, S. R., Saleh, G. A., Mohamed, F.
TLC Videodensitometer software were shown to
A., and Rageh, A.H. (2007) Analysis of
be comparable to TLC-densitometry. From data
Cephalosporins. Journal of Pharmaceutical
above, TLC-image analysis using Sorbfil TLC
and Biomedical Analysis 45: 1-19.
Videodensitometer could be further applied for
Eric-Jovanovic, S., Agbaba, D., Zivanov-Stakic,
rapid determination of CFX in bulk and dosage
D., and Vladimirov, S. (1998) HPTLC
forms and might be used as alternative to more
determination of ceftriaxone, cefixime and
expensive quantitative chromatographic methods,
cefotaxime in dosage forms. Journal of
e.g., TLC densitometry and HPLC, which could not
Pharmaceutical and Biomedical Analysis
be afforded by small labs.
18: 893-898. Gáspár, A., Kardos, Sz., Andrási, M., and Klekner,
Acknowledgements
Á. (2002) Capillary electrophoresis for
The authors wish to thank Silpakorn University
the direct determination of cephalosporins
Research and Development Institute, Thailand, for
in clinical samples. Chromatographia
the financial support of this research.
Supplement 56: 109-114. Hung, T. T. N., Dinh, C. P., and Duc, C. H. (2001)
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13
Research Article Differentiation of a Hyperthermophilic Archaeon Pyrococcus sp. strain Pikanate 5017, by Arbitrarily Primed PCR Patlada Pasomsup1, Juan Miguel González2, Maria Carmen Portillo2, Veeranun Pongsapukdee3 and Wirojne Kanoksilapatham1* 1
Department of Microbiology, Faculty of Science, Silpakorn University, Nakhon Pathom, Thailand 2 IRNAS-CSIC Avda. Reina Mercedes 10, 41012 Sevilla, Spain 3 Department of Statistics, Faculty of Science, Silpakorn University, Nakhon Pathom, Thailand * Corresponding author. E-mail address: wirojne@su.ac.th Received April 21, 2010; Accepted March 23, 2011
Abstract Pyrococcus and Thermococcus are hyperthermophilic archaea in the order Thermococcales. Both genera are strictly anaerobes, gaining energy by fermentation of peptide and several carbohydrates at optimal temperature above 70 °C. A Pyrococcus sp. strain Pikanate 5017 (PK 5017) was recently isolated from a hot spring in Northern Thailand. The strain PK 5017 is a valuable source of numerous genes encoding thermostable enzymes. Growth kinetics determined at various temperatures (75-105 °C) indicates that strain PK5017 is a fast growing archaeon. An arbitrarily primed PCR (AP-PCR) technique was successfully applied to differentiate the genome sequences of six members of Thermococcales. Two single primers, ARB-1f (5’ ATGAG GACT GAAA CCATT 3’) and ARB-2f (5’ GTAAA ACGA CGGC CAGT 3’), are effective in producing polymorphisms of the PCR products at 3-10 ng of DNAs. The unique AP-PCR fingerprints distinguish the strain PK5017 from P. furiosus, P. horikoshii, P. abyssi, T. litoralis, and T. celer. The results indicate a clear distinction of genome sequences among Pyrococcus and Thermococcus genera. Key Words: Thermococcales; Pyrococcus; Thermococcus; fingerprint; hyperthermophile; hot spring
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Introduction Arbitrarily primed PCR (AP-PCR) is a PCR based technique employing a single primer to amplify PCR products, representing fingerprints of genomes of organisms (Welsh and McClelland,
celer (Zillig et al., 1983), T. litoralis (Neuner et al., 1990), T. profundus (Kobayashi et al., 1994), T. chitonophagus (Huber et al., 1995), T. thioreducens (Pikuta et al., 2007), Pyrococcus furiosus (Fiala and Stetter, 1986), P. woesei (Zillig et al., 1987), P. abyssi (Erauso et al., 1993), P. horikoshii (González et al., 1998), and P. glycovorans (Barbier et al., 1999). However, freshwater species have been restricted to the genus Thermococcus, including T. zilligii (Ronimus et al., 1997), T. waiotapuensis (González et al., 1999), and T. sibilicus (Miroshnichenko et al., 2001). Among the marine Pyrococcus species, P. furiosus and P. woesei are two closely related species which were isolated from a shallow hot vent of Vulcano Island, Italy. The latter species was renamed as P. furiosus subsp. woesei. Both P. furiosus and P. woesei have identical 16S rRNA gene operon, highly similar genome sequences, and several copies of the homologous insertion sequence of elements in many identical loci of their genomes (Kanoksilapatham et al., 2004). P. abyssi, P. horikoshii, and P. glycovorans were isolated from deep-sea hydrothermal vents of the North Fiji Basin in the South West Pacific Ocean, the Okinawa Trough in the North East Pacific Ocean, and the East Pacific Rise, respectively. A Pyrococcus sp. strain HT3 grew optimally at 80-85 °C, pH 7.5 and 1.5% NaCl was isolated from a hot spring in Northeast Algeria (Kecha et al., 2007). However, the optimal growth temperature reported is in the range of Thermococcus (below 90 °C). Pyrococcus sp. strain Pikanate 5017 was isolated from a fresh water hot spring in Northern Thailand (unpublished data). This study has the objectives of 1) comparing various genome sequences of the members in the Order Thermococcales by using an AP-PCR technique with two arbitrarily single primers; and 2) characterizing growth kinetics on the temperature range of the strain Pikanate 5017.
1990; Welsh et al., 1991; Keller et al., 1995). The technique has a number of applications in epidemiology, ecology, taxonomy, and organism identification. The technique is flexible because nucleotide sequences and lengths of the primers can be arbitrarily chosen and performed without prior sequence information. An oligonucleotide sequence of 5’ GTAAA ACGA CGGC CAGT 3’ was shown to be effective in distinguishing a marine hyperthermophilic archaeon, Thermococcus alcaliphilus, from its relatives (Keller et al., 1995). Hyperthermophiles in the order Thermococcales are anaerobic, heterotrophic Archaea belonging to Euryarchaeota (Woese et al., 1990). Pyrococcus and Thermococcus are two closely related genera of hyperthermophilic archaea in the order Thermococcales. Both grow anaerobically by fermenting peptides and carbohydrates at optimal temperatures above 70 °C (Adams, 1994; Stetter, 1996). Pyrococcus species grow optimally above 90 °C. However, Thermococcus species grow optimally below 90 ° C (Barbier et al., 1999; González et al., 1999). Both genera have highly homologous 16S rRNA gene sequences, and can be distinguished using their optimal growth temperatures, physiological characteristics, and DNA properties. The typical ecological systems for the habitats of both genera include deep and shallow hydrothermal vents, geothermal subterranean and hot springs (Stetter, 1993; Zillig et al., 1983; Pikuta et al., 2007). Most members of Thermococcales are marine isolates and require approximately 3% (w/v) of NaCl for their optimal growth, including Thermococcus
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Materials and Methods Organisms Pyrococcus sp. strain Pikanate 5017 was recently isolated from Pong Dueat Hot Spring (unpublished data). Microorganisms used in this study were P. abyssi (strain GE5; Erauso et al. 1993), Pyrococcus furiosus (DSM3638), P. horikoshii (DSM12428), Thermococcus celer (DSM2476), and T. litoralis (DSM5473). Cultivation Medium is composed of (per liter) 24 g NaCl, 4 g Na2SO4, 0.7 g KCl, 0.2 g NaHCO3, 0.1 g KBr, 0.03 g H3BO3, 10.8 g MgCl2.6H2O, 1.5 g CaCl2.2H2O, 0.025 g SrCl2.6H2O, 5 g tryptone, 1 g yeast extract, 1 ml resazurin solution (0.2 g l–1), 3 ml Na2S.9H2O solution [25% (w/v), pH 7], and 5 10 g elemental sulfur. Cultures were inoculated in serum bottles containing 100 ml of the medium, and incubated anaerobically at appropriate temperatures (95 °C for Pyrococcus, and 85 °C for Thermococcus). Phylogenetic study The 16S rRNA gene sequence (total of 1330 bp long) from strain PK 5017 (GenBank accession number FJ793195 and HQ223090) and reference species (see below) were aligned by Neighbor joining algorithms using the program Clustal W2-Multiple Sequences Alignment (http://www.ebi.ac.uk/Tools/ msa/clustalw2/). The GenBank accession numbers of the 16S rRNA sequences from the reference species used in this study are as follows: Pyrococcus furiosus (U20163), P. horikoshii (D87344), P. abyssi (L19921), P. glycovorans strain AL585 (AY099168), Thermococcus litoralis (AY099180), T. celer (M21529), T. fumicolans (Z70250), T. hydrothermalis (Z70244), T. profundus (Z75233), T. sibiricus (AJ238992), T. zilligii (U76534), and Palaeococcus ferrophilus (AB019239). The last number belongs to the sequence of the genus in the order Thermococcales (Takai, et al., 2000). An unrooted phylogenetic tree was constructed by setting a bootstrap value of 1000, and the 16S rRNA sequence of Sulfolobus solfataricus (GenBank
accession number D26490) was employed as outgroup. Determination of growth parameters Cultures were grown in 100 ml serum bottles, and samples (1 ml) were drawn at one-hour time interval between 0-6 h. Cell numbers were counted using the MPN method. Specific growth rates (m) were estimated based on the regression line obtained from plotting ln N versus t: N is the number of cells ml–1 and t is the incubation time (h). At least four data points were used for the regression analyses during the exponential phase of growth. Genomic DNA preparation DNA was extracted from 300 ml of overnight cultures following the method described by Charbonnier and Forterre (1995). Briefly, cell pellets were suspended in 400 µl TNE buffer (10 mM Tris, pH 8.0; 100 mM NaCl; 1 mM EDTA) in a microcentrifuge tube. Fifty microliters of 20% N-lauroyl sarcosine and 100 µl of 10% SDS were added. After inverted mixing, 100 µl of 10 mg/ ml Proteinase K solution was added and incubated at 50 °C for a few hours (or overnight). Proteins were precipitated twice by equal volume (750 µl) of TE-saturated phenol. The aqueous (top layer) phase was separated by centrifugation at 5,000x g for 10 minutes at room temperature, and was subsequently extracted twice using an equal volume (750 µl) of chloroform/isoamyl alcohol (24:1) for approximately 10 minutes. The upper phase was collected by centrifugation (5,000x g for 10 minutes) followed by precipitation of genomic DNA using cold absolute ethanol (-20 °C) in a refrigerator. DNA pellet was collected and washed at least three times at room temperature using 70 % (v/v) ethanol. The dried DNA pellet was dissolved in 50-100 ml of TE buffer (10 mM Tris, pH 8.0; 1 mM EDTA). RNAs in the sample were digested using DNAse-free RNAse (10 mg/ml in TE buffer) at 37 °C for at least one hour. The DNA solution was then subjected to ethanol precipitation one more time. DNA solution was stored at 4 °C. DNA concentration (ng/ml) was 16
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polymerization was completed at 72 °C for 10 min. The reactions were stored at 4 °C. The amplified products were separated using a conventional 1% agarose gel electrophoresis.
determined using a spectrophotometric method. Arbitrarily-Primed PCR Reactions Synthetic oligonucleotide primers, named in this study as “ARB-1f” and “ARB-2f” were from Pacific Science Co., Ltd., Singapore. The nucleotide sequence of the ARB-1f primer (5’ ATG AGG ACT GAA ACC ATT 3’) was chosen because it was likely to bind to the 5’ end of a highly homologous transposase-integrase gene family, which is present in multiple copies in the genomes of P. furiosus (Kanoksilapatham et al., 2004). The nucleotide sequence of the ARB-2f primer (5´ GTA AAA CGA CGG CCA GT 3´) was described elsewhere (Keller et al., 1995). A 50 ml reaction volume contained genomic DNAs (3-10 ng), 10 x Taq buffer (5 ml), a primer (20 pmol), dNTP (10 nmol), 50 mM MgCl2 (1.5 ml), and Taq DNA polymerase (2.5 units). PCR was performed essentially as described by Welsh and McClelland (1990). Briefly, to produce long DNA templates, two cycles of a reaction condition of 94°C for 5 min, 40°C for 5 min, and 72°C for 5 min were conducted. Then, the polymorphic DNA templates obtained were amplified, using 40 cycles of a standard PCR reaction conditions (94 °C for 1 min, 40 °C for 1 min, and 72 °C for 2 min). The
Results Pyrococcus sp. strain Pikanate 5017 Pyrococcus sp. strain Pikanate 5017 (PK5017) was isolated from a sediment sample collected from a geyser type hot spring located at Huai Nam Dang National Park (19° 7′ 19″ N, 98° 56′ 37″ E), Chiang Mai Province, Thailand (unpublished data). Strain PK5017 is a strict anaerobe growing on peptides as sole carbon and energy sources (see medium compositions in Materials and Methods). 16S rRNA sequences of the strain PK 5017 (GenBank Accession Numbers FJ793195 and HQ223090) were determined and deposited to the NCBI public database. Phylogenetic analysis reveals that PK 5017, P. abyssi, P. horikoshii and P. furiosus are closely related species (Figure 1). In contrast, P. glycovorans, Palaeococcus ferophilus (a member in the order Thermococcales) and the other Thermococcus species are grouped together.
Figure 1 Phylogenetic tree of PK 5017 and some members of Thermococcales. PK 5017 is classified in the Pyrococcus lineages. 17
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Growth kinetics Growth kinetics of strain PK 5017 is assayed in exponential phase of growth (Figures 2a and 2b). Figures 2a and 2b indicates that an optimal temperature for growth is 95 °C, with a mmax value of 2.67 h-1 (R2 = 0.90). Growth rate at 105 °C is approximately 2.5 times slower, with an estimated doubling time of 40 min (R2 = 0.92). Minimal
growth temperature is predicted to be lower than 75 °C, because a positive mmax value of 0.85 h-1 (R2 = 0.99) is measured. Specific growth rate constants calculated at 80, 85, 90, 100 and 105 °C are 0.89 h-1 (R2=0.87), 1.25 h-1 (R2=0.87), 2.48 h-1 (R2 = 0.95), 2.48 h-1 (R2 = 0.95), 1.78 h-1 (R2=0.66) and 1.03 h-1 (R2=0.92), respectively.
Figure 2 Growth kinetics of Pyrococcus sp. strain Pikanate 5017. Cell count was performed using MPN method (see also Materials and Methods). (a) Specific growth rates (h–1) were estimated based on the regression lines obtained from plotting log cells ml-1 (y axis) versus incubation time in hours (x axis). Optimal growth temperature is 95 °C (mmax = 2.67 h-1; R2 = 0.90). Maximal temperature is 105 °C (mmax = 1.03 h-1; R2 = 0.92). Symbols: solid diamond represents exponential growth at 75 °C; solid square represents exponential growth at 80 °C; solid triangle represents exponential growth at 85 °C; solid circle represents exponential growth at 90 °C; blank square represents exponential growth at 95 °C; blank circle represents exponential growth at 100 °C; blank triangle represents exponential growth at 105 °C. (b) Reciprocal values of the specific growth rate (in Figure 2a) were plotted versus temperatures. An optimal growth temperature range of 90-95 °C is predicted.
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Because optimal temperature for growth of the strain PK 5017 is above 90 °C (Figures 2a and 2b), 16S rRNA sequence (GenBank Accession Number FJ793195) is highly related to sequence belonging to species in the order Thermococcales, the strain PK 5017 was classified to be a member of the genus Pyrococcus (Figure 1). However, comparisons of genome sequences (an essential parameter for species differentiation) have never been described. Reliabilities of AP-PCR Technique In order to differentiate genome sequences, two single primers were arbitrarily chosen and named in this study as “ARB-1f” and “ARB-2f” (see nucleotide sequences in Materials and Methods). Initially, reliability of the primers was tested on varied genomic DNA concentrations of 3-10 ng. A concentration of each single primer was fixed at 20 pmol in a 50 ml reaction volume. Results are shown in Figure 3. Figure 3a reveals consistent patterns obtained from DNAs of Pyrococcus furiosus (lanes 1 and 2), and strain PK5017 (lanes 3 and 4) performed at concentration of 5 and 10 ng, respectively. Similarly, a distinct pattern was obtained when the template DNA (3, 5 and 10 ng) of P. abyssi (lanes 1-3 in Figure 3b) was employed. The results imply that the single ARB-1f primer is reliable in generating PCR profiles regardless of the template concentrations tested. Comparisons among the patterns reveal unique polymorphisms. However, a common band with an approximate size of 700 bp-long (indicated by an arrow in Figure 3a) was observed on the templates of P. furiosus and strain PK5017, indicating sequence similarity in a region(s) of these archaea. Polymorphisms of AP-PCR products generated using the ARB-2f primer are shown in Figures 3c - 3e. As expected, the results reveal three consistent patterns of strain PK5017 (Figure 3c), P. furiosus (Figure 3d), and P. abyssi (Figure 3e), respectively, suggesting reproducibility of the technique regardless of concentrations and sequence of the primer tested. In addition, clearly different
patterns obtained using the ARB-2f confirms different genome sequences among the three hyperthermophilic archaea. Genomic Fingerprints of Thermococcales In order to compare genomic fingerprints, AP-PCR reactions were conducted using either the ARB-1f or ARB-2f primer with 5 ng of templates. Results are shown in Figure 4. Figure 4a reveals different AP-PCR patterns generated by the ARB1f primer. The patterns obtained are varied in band numbers and sizes, indicating different genome sequences in several regions. The profiles of the Pyrococcus abyssi (lane 3), P. horikoshii (lane 4), and Thermococcus celer (lane 6) are unique, and contain no prominent band in common. However, the band sized of 700 bp-long mentioned in previous section (Figure 3a) was also observed in the pattern of Thermococcus litoralis (lane 5), indicating sequence similarity of T. litoralis to the strain PK 5017 (lane 1) and P. furiosus (lane 2). The results shown in Figure 4b confirm unique profiles when the ARB-2f was employed. In Figure 4b, the patterns of the strain PK 5017 (lane 1) and P. furiosus (lane 2) are not identical. However, at least 5 bands (indicated by arrows in Figure 4b), were commonly observed in both species. The band patterns of P. abyssi (lane 3), P. horrikoshii (lane 4), T. celer (lane 5), and T. litoralis (lane 6) are clearly distinguishable from the strain PK5017. Discussion Hyperthermophiles have adapted themselves to thrive in habitats with extremely high temperature, and are important sources of highly thermostable enzymes (Stetter et al., 1993). Hyperthermophilic archaea in the order Thermococcales have been isolated from hydrothermal vents mostly in marine solfatara. Pyrococcus sp. strain Pikanate 5017 (PK5017) was first isolated from a terrestrial and sulfide rich hot spring in Northern Thailand (unpublished data). Growth parameters determined between the temperature range of 75-105 °C (Figure 19
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Figure 3 Reproducibility of AP-PCR technique at varied template concentrations. (a) DNAs were amplified using the ARB-1f primer (5’ ATG AGG ACT GAA ACC ATT 3’), templates of P. furiosus (lanes 1 and 2), and strain PK5017 (lanes 3 and 4). Lanes 1 and 3 represent 5 ng of the templates. Lanes 2 and 4 represent 10 ng of the templates. (b) DNA was amplified from a template of P. abyssi using ARB-1f primer. Lanes 1-3 represent 3, 5 and 10 ng of the template concentrations, respectively. (c) DNA was amplified from a template of strain PK5017 using ARB-2f primer (5´ GTA AAA CGA CGG CCA GT 3´). Lanes 1-3 represent 3, 5 and 10 ng of the template concentrations, respectively. (d) DNA was amplified from a template of P. furiosus using ARB-2f primer. Lanes 1-3 represent 3, 5 and 10 ng of the template concentrations, respectively. (e) DNA was amplified from a template of P. abyssi using ARB-2f primer. Lanes 1-3 represent 3, 5 and 10 ng of the template concentrations, respectively. Sizes of markers (bp) are indicated on the left.
and was the isolation temperature. The 16S rRNA partial sequence of the strain PK 5017 (GenBank Accession Numbers FJ793195 and HQ223090) are very similar (96-100 %) to the 16S rRNA gene sequences of most Thermococcales (data not shown). Phylogenetic analysis grouped the strain PK 5017 with Pyrococcus (Figure 1). However, the 16S rRNA gene sequences do not allow the accurate differentiation of species and strains within the genera Pyrococcus and Thermococcus. An AP-PCR technique is successfully applied to explore genomic fingerprints, one of the essential
2) indicate a fast growing hyperthermophile calculated to be less than 20 min per generation (mmax = 2.67 h-1; R2 = 0.90) at optimal conditions (95 °C, 2.4% NaCl, pH 7.2). Because its optimal temperature for growth is above 90 °C, the strain PK5017 is classified to be a member of the genus Pyrococcus (optimal temperature of Thermococcus is less than 90 °C; Barbier et al., 1999; González et al., 1999). It is observed that the mmax value (2.48 h-1; R2 = 0.95) determined at 90 °C is near the value obtained at 95 °C. This temperature (90 °C) is close to temperature of the sediments measured in situ 20
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Figure 4 Genomic fingerprints of Thermococcales at 5 ng of genomic DNAs. (a). AP-PCR polymorphisms were generated using the ARB-1f primer (5’ ATG AGG ACT GAA ACC ATT 3’). Lanes 1-6 represent templates of Pyrococcus sp. strain PK5017, P. furiosus, P. abyssi, P. horikoshii, Thermococcus litoralis, and T. celer, respectively. (b). AP-PCR polymorphisms were generated using ARB-2f primer (5´ GTA AAA CGA CGG CCA GT 3´). M represents size markers. Lanes 1-6 represent templates from Pyrococcus sp. strain PK5017, Pyrococcus furiosus, P. abyssi, P. horikoshii, Thermococcus celer, and T. litoralis, respectively. The distinguishable patterns suggest different genome sequences. Conclusions Simplicity and flexibility of the AP-PCR technique is clearly demonstrated in this paper because the technique requires no prior sequence information and can be conducted in a single step with slight modification of a standard PCR reaction. Reliability is disclosed at a DNA concentration range of 3-10 ng (Figures 3). Two arbitrarily primers named in this study as “ARB-1f” (5’ ATG AGG ACT GAA ACC ATT 3’) and “ARB-2f” (5´ GTA AAA CGA CGG CCA GT 3´) were successfully employed to compare the genome sequences of the hyperthermophilic Pyrococcus sp. strain Pikanate 5017 with those of the closely related species. The growth parameters determined in this study help cell mass optimization and thermostable enzyme production. The strain PK 5017 can be considered as a source of numerous genes, encoding extremely heat stable enzymes. Clearly different fingerprints suggest that strain PK5017 is a distinct species within the genus Pyrococcus.
DNA parameters, of six representative members of the Archaea within the Thermococcales. Reliability is disclosed at a genomic DNA concentration range of 3-10 ng (Figures 3). The bands sized of 700 bplong (compare lanes 1 and 2 with lanes 3 and 4 in Figure 3a), and several bands (compare lane 1 with lane 2 in Figure 4a) observed in common between the strain PK5017 and P. furiosus, suggesting that the strain PK5017, compared with the others, is closely related to P. furiosus. The unique AP-PCR fingerprints (Figures 4a and 4b) distinguish the strain PK 5017 from P. furiosus DSM3638, P. horikoshii strain OT3, P. abyssi strain GE5, T. litoralis, and T. celer. The results generated from the use of APPCR technique, unlike 16S rRNA gene sequence analysis, indicate a clear distinction of species and strains belonging to the genera Pyrococcus and Thermococcus.
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Acknowledgements This work was supported by a grant from the Scientific Promotion and Development Fund, Faculty of Science, Silpakorn University (Grant number RGI 2553-06). JMG and MCP acknowledge the support from a CSIC movility program, PA1001993 and PA1002058, and the Andalusian Government Bio288. Our appreciation goes to Prof. Frank T. Robb (University of Maryland Biotechnology Institute, USA) for his collaboration.
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Research Article All Congruence Modular Symmetric and Near-Symmetric Algebras Chawewan Ratanaprasert1* and Supharat Thiranantanakorn1 Department of Mathematics, Faculty of Science, Silpakorn University, Nakhon Pathom, Thailand * Corresponding author. E-mail address: ratach@su.ac.th
1
Received February 11, 2011; Accepted April 25, 2011
Abstract
For a unary operation f on a finite set A , let denote l ( f ) the least non-negative integer with
Im f l ( f ) = Im f l ( f ) +1 which is called the pre-period of f . K. Denecke and S. L. Wismath have characterized all operations f on A with l ( f ) = A − 1 and prove that l ( f ) = A − 1 if and only if there exists a d ∈ A such that A = {d , f (d ), f 2 (d ), , f
A −1
(d )} where f
A −1
(d ) = f A (d ) . C. Ratanaprasert and K. Denecke
have characterized all operations f on A with l ( f ) = | A | −2 for all | A | ≥ 3 ; and have characterized all equivalence relations on A which are invariant under f with these long pre-periods.
In the paper, we study finite unary algebras A = ( A ; f ) with l ( f ) ∈ {0, 1} for | A | ≥ 3
which are called symmetric algebras and near-symmetric algebras, respectively. We characterize all operations f whose A is congruence modular. We prove that a symmetric algebra A is congruence modular if and only if the lattice ConA of all congruence relations is either a product of chains or a linear sum of a product of chains with one element top or a M 3 − head lattice; and a near-symmetric algebra A is congruence modular if and only if ConA is one of the followings: 2 × P, 2 × ( P ⊕ 1), 2 × L, M 3 × P, M 3 × ( P ⊕ 1), or M 3 × L where P denote a product of chains and L is a M 3 − head lattice. Key Words: Monounary algebra; Congruence distributive; Congruence modular
Silpakorn U Science & Tech J 5 (1) : 24-33, 2011
C. Ratanaprasert and S. Thiranantanakorn
Silpakorn U Science & Tech J Vol.5(1), 2011
Introduction
for | A | ≥ 3 and they also have characterized all
Monounary algebras are algabras with one
equivalence relations on A which are invariant
unary fundamental operation. Mono-unary and
under f with l ( f ) = | A | −1 for | A | ≥ 2 and
partial monounary algebras play a significant role
l ( f ) = | A | −2 for | A | ≥ 3 . These answer the
in the study of algebraic structures. Moreover,
above open problem for some of monounary
there exists a close connection between monounary
algebras. Besides, the results convince us that the
algebras and some types of automata. The advantage
pre-period of unary functions defined on a finite
of monounary algebras is their relatively simple
set will be a kind of notions for classifications of
visualization. They can be represented by a graph,
finite algebras. At the beginning of the eighties,
which is always planar, hence easy to draw. Unary
R. McKenzie and D. Hobby (1998) developed a
algebras were first intensively investigated by
new theory, called “Tame Congruence Theory”
B. Johnson about 40 years ago and were investigated
which offers a structure theory for finite algebras.
mainly by D. J. Studenovska (1982, 1983). The
If Im f = A or Im f = 1 , then A = ( A ; f ) is
problem of describing the lattices which are
called a permutation algebra; that is, l ( f ) ∈ {0, 1}
isomorphic to congruence lattices of monounary
and f is of short pre-periods. Permutation algebras
algebras is still open.
play an important role in tame congruence theory.
Let A be a finite set and denote by | A | ≥ 2
the cardinality of A . For a unary operation f on
In this paper, we are interested in formulating
a characterization of all unary operations defined
A, let Im f = { f ( x) x ∈ A} be the image of f and
on a finite set A with short pre-periods. We prove
let l ( f ) be the least non-negative integer m such
necessary and sufficient conditions of f whose
that Im f m = Im f m +1 . The number l ( f ) is called
permutation algebras are congruence modular and
the pre-period of f , sometimes also the stabilizer
then characterize all lattices which are isomorphic
of f . K. Denecke and S.L. Wismath (2002) have
to the congruence lattice of modular permutation
proved that l ( f ) = | A | −1 if and only if there
algebras.
exists a d ∈ A such that A = {d , f (d ), f 2 (d ),
, f
A −1
All lattices which are the congruence lattices of
(d )} which shows a characterization of
all longest pre-periods f .
modular symmetric algebras.
It is well-known that the congruence lattice
In this section, we assume that A is a
of an algebra is uniquely determined by the unary
finite set and f is a unary operation on A with
polynomial operations of the algebra. C. Ratanaprasert
l ( f ) = 0 . Note that l ( f ) = 0 if and only if f
and K. Denecke (2008) have characterized all unary
is a permutation on A . In the theory of groups, the
operations f on a finite set A with l ( f ) = | A | −2
group of all permutations on a nonvoid set A is
25
Silpakorn U Science & Tech J Vol.5(1), 2011
All Congruence Modular Symmetric and Near-Symmetric Algebras
known as a symmetric group and it is well known
contains θ j for all j ∈ {1, 2,3}; and if (a, b) ∈ θ
that every permutation can be decomposed into
then {a, b} ⊆ B j ∪ Bσ ( j ) for some j ∈ {1,2,3}
simple parts called cycles. We call the monounary
; hence, (a, b) ∈ θ j ∨ θσ ( j ) ; so, θ j ∨ θσ ( j ) = θ
algebra A = ( A ; f ) in this case, a symmetric
for all j ∈ {1,2,3} . Therefore, ConA is not
algebra. If A is a singleton or a two-elements set,
distributive since it contains a M 3 − sublattice
the lattice ConA of all congruence relations on A
{θ1 ,θ2 ,θ3 , ω ,θ } .
is also a singleton set {D A } or a two-elements chain
{D A , A × A} , respectively; so, A is congruence-
distributive then ConA is the modular lattice M 3 .
distributive. We are interested in the case | A | ≥ 2 .
Remark 1. Let ( B ; f B ) be a subalgebra of A .
We first consider necessary conditions for f whose
We will denote the restriction θ |B of θ ⊆ A × A
A is congruence-distributive.
on B by θ B . And if θ ⊆ B × B, we will denote
¨
We note that if A = 3 and ConA is not
the relation θ ∪ {( x, x) | x ∈ A} by
Recall that the notation θ ( B ) stands for the
θ A . Then
least congruence on an algebra A which contains
θ B ∈ Con( B; f B )
a subset B of A .
θ A ∈ ConA for all θ ∈ Con( B ; f B ) .
for
Remark 2. Let
all
θ ∈ ConA
(a f (a) f
p−1
(a)
and
and
Proposition 1. If A is a congruence distributive
(b f (b) f q−1 (b) be cycles in the product of f
symmetric algebra with | A | ≥ 3 then either f is
for some positive integers p and q and θ ∈ ConA .
a cycle having at most one fixed point or, f has
no fixed points and f is a product of two disjoint
then (a, f
cycles whose lengths are relatively prime.
Proof. Suppose that f = a1a 2 a r for r ≥ 3 and
0 < r ≤ p − 1 which is not a factor of p then
a i and a j are disjoint cycles for all 1 ≤ i ≠ j ≤ r .
{ a, f (a ), , f p −1 (a )} is contained in a block of
(i) If (a, f r (a ) ∈ θ for some 0 < r ≤ p − 1 , kr
(a ) ∈ θ for all non-negative integer k .
(ii) If (a, f r (a ) ∈ θ for some integer
Let σ = (123) and denote by Bi , the set of all
the quotient algebra A .
elements in the cycle a i for all 1 ≤ i ≤ r . Since
Bi ∩ B j is empty and f ( x) ∈ Bi whenever
{a, f (a ), , f
x ∈ Bi for all 1 ≤ i ≤ r , the relations θ j := D A ∪
contained in a block of A .
{ ( x, y) |{x, y} ⊆ B ∪ B j
σ ( j)
θ
(iii) If ( p, q ) = 1 and (a, b) ∈ θ , then p −1
(a ), b, f (b), , f q −1 (b)} is θ
}
or {x, y} ⊆ Bσ 2 ( j ) is
Recall that a linear sum of an ordered set P
ω := θ j ∧ θσ ( j ) =
with a one-element chain 1 is an ordered set P ⊕ 1
D A ∪ {( x, y ) | x, y ∈ Bk } for all j ∈ {1, 2,3} .
which can represents P with a new top element
invariant 3
under f and
k =1
Also, the congruence θ = θ ( B1 ∪ B2 ∪ B3 )
added.
26
C. Ratanaprasert and S. Thiranantanakorn
Silpakorn U Science & Tech J Vol.5(1), 2011
Proposition 2. If f satisfies the conditions of
so, θ1A ∨ θ 2A ≠ A × A;
Proposition 1 then ConA is either a product of
Now, if θ ∈ ConA \{ A × A} then θ BA1 ∪ θ BA2
hence,
A × A ∉ Im β .
chains or a linear sum of a product of chains with a
∈ ConA where
one-element chain.
with a ∈ B1 and b ∈ B2 , the result (iii) in
Proof. Let A = n and f be a cycle having no fixed
Remark 2 implies that A = B1 ∪ B2 is a subset of
point and a ∈ A . Let denote ↓n the lattice of all
a block of A
θ BA ∪ θ BA ⊆ θ . If (a, b) ∈ θ 1
2
factors of n ordered by the division of integers. For each
θ since the lengths of a1 and a 2 are relatively prime; so, θ = A × A , a contradiction.
m ∈↓ n ,let f [ j ] (a) = { f s (a) | s ≡ j (mod m)} .
So, if (a, b) ∈ θ then {a, b} ⊆ Bi for some
Then ℘m := f [ j ]m (a ) | j = 0,1, 2, , m − 1 is a
i ∈ {1, 2} ; hence, (a, b) ∈ θ Bi ⊆ θ BA1 ∪ θ BA2 . Thus,
partition of A which corresponds to the congruence
θ ∈ Im β . Therefore, ConA \{ A × A} = Im β
θ m modulo m restriction to A ; that is, θ m =
≅ Con( B1 ; f |B1 ) × Con( B2 ; f |B2 ) . Hence, ConA
m
{
{( x, y) | x, y ∈ f
}
[ j ]m
}
(a ) for j = 0,1, 2, , m − 1 .
is a linear sum of a product of chains P and a one-
Hence, the map a :↓ n → ConA defined by
element chain 1 .
a (m) = θ m for all m ∈↓ n is clearly an order-
isomorphism; so, ConA is dually isomorphic to
distributive if A is singleton or a two-elements set.
↓n which is a product of chains.
In the case | A | = 3 , if f is identity then ConA
¨
We knew that an algebra A is congruence-
Next, assume that f = a1a 2 where a1 and a 2
is modular; and if f is not identity, Proposition 2
are disjoint cycles whose lengths are relatively prime
implies that ConA is distributive. We will consider
whenever both of them are of lengths more than one.
the case | A | ≥ 4.
Then, f | Bi is a cycle on the set Bi of all elements in the cycle a i for i ∈ {1,2} . Hence, Con( Bi ; f | Bi ) is
Proposition 3. If a symmetric algebra A with
a product of chains for i ∈ {1,2} . Since {B1 , B2 } is
| A | ≥ 4 is congruence-modular, then f is either
a the partition on A, θ1A ∪ θ 2A is a congruence on
one of the followings:
A for all θi ∈ Con( Bi ; f | Bi ) and i ∈ {1, 2}; hence,
ConA is a sublattice of the power set of A × A, ;
points, or
so, the map β : (θ1 , θ 2 ) → θ1A ∨ θ 2A is an order
embedding from Con( B1 ; f |B1 ) × Con( B2 ; f |B2 )
product of two disjoint cycles whose lengths are
into ConA .
relatively prime, or
Let θ i ∈ Con( Bi ; f | Bi ) for i ∈ {1,2} . Then
( x, y ) ∉ θ1A ∨ θ 2A whenever x ∈ B1 and y ∈ B2 ;
(i) f is a cycle having at most two fixed
(ii) f has at most one fixed point and f is a
(iii) f has no fixed point and f is a product
of three disjoint cycles whose lengths are relatively
27
Silpakorn U Science & Tech J Vol.5(1), 2011
All Congruence Modular Symmetric and Near-Symmetric Algebras
prime.
all k ∈ Bi ; hence, (k , l ) ∈ θ1 for all k ∈ Bi and
Proof. Similar arguing as the proof of Proposition
l ∈ Bσ (i ) ; and also, (k , s ) ∈ θ1 for all k ∈ Bi
1. Suppose that f is a product of at least four
and s ∈ Bσ (i ) which implies that ( s, m) ∈ θ 3 for
disjoint cycles a1a 2 a r where r ≥ 4 and all a i
all m ∈ Bσ 3 ( i ) and s ∈ Bσ (i ) ; thus, ( s, qk ) ∈ θ 3
and a j are disjoint cycles (can be of length 1) for
for all s ∈ Bσ (i ) . Now, a θ 3 x θ1 s θ 3 qk θ 3 b
1 ≤ i ≠ j ≤ r . Then, we construct the following 3
implies that ( a, b) ∈ θ1 ∨ θ 3 . If (q j , q j +1 ) ∉ θ 3
congruences :
for all k ≤ j ≤ t − 1 , the same arguing as above
θ1 = D A ∪ {( x, y ) |{x, y} ⊆ Bi ∪ Bσ (i ) or {x, y} ⊆ Bσ
2
θ 2 = D A ∪ {( x, y ) |{x, y} ⊆ Bi ∪ Bσ (i ) or {x, y} ⊆ Bσ
θ3 = D A ∪ {( x, y ) |{x, y} ⊆ Bi ∪ Bσ
2
(i )
and
(i )
∪ Bσ 3 (i ) } and
and by continuing the process one can prove that
4
{( x, y) | x, y ∈ B } k =1
2
or {x, y} ⊆ Bσ (i ) ∪ Bσ 3 (i ) }
Where σ = (1234) . One can show that θ1 ⊆ θ 2 ,
θ1 ∧ θ3 = θ 2 ∧ θ3 = D A ∪
or {x, y} ⊆ Bσ 3 (i ) }
(i )
θ 2 ∨ θ 3 ⊆ θ1 ∨ θ 3 . So, θ1 ∨ θ 3 = θ 2 ∨ θ 3 .
k
θ1 ∨ θ 3 ⊆ θ 2 ∨ θ 3 . If (a, b) ∈ θ 2 ∨ θ 3 ,
Therefore, ConA is not modular since it contains
(qk , qk +1 ) ∈ θ 2 ∪ θ 3 for all 0 ≤ k ≤ t − 1 . We may
a N 5 − sublattice {θ1 , θ 2 , θ 3 , θ1 ∧ θ 3 , θ1 ∨ θ 3 } . ¨
assume that (a, q1 ) ∈ θ 2 . The finiteness of the set
Q = {a = q0 , q1 , , qt = b} implies the existence of
modular if | A | ≥ 4 and f is an identity on A .
there are a = q0 , q1 , , qt = b ∈ A such that
the greatest element qk ∈ Q such that (qk , qk +1 ) ∈ θ 3
We note from Proposition 3 that ConA is not
but (qi −1 , qi ) ∈ θ 2 for all 1 ≤ i ≤ k ; so, (a, qk ) ∈ θ 2 .
Proposition 4. If A is a symmetric algebra with
If (q j , q j +1 ) ∈ θ 3 for each k ≤ j ≤ t − 1 then
| A | ≥ 4 whose f has no fixed points and f is a
(qk , b) ∈ θ 3 ; hence, {a, qk } ⊆ Bi ∪ Bσ (i ) or
product of three disjoint cycles all of them are of
{a, qk } ⊆ Bσ 2 (i ) ∪ Bσ 3 (i ) . If {a, qk } ⊆ Bi ∪ Bσ (i )
relatively prime lengths, then ConA is modular
then (a, qk ) ∈ θ1 ; so, ( a, b) ∈ θ1 ∨ θ 3 ; but,
which is not distributive.
if {a, qk } ⊆ Bσ 2 ( i )
then
Proof. Let f = a1a 2a 3 satisfy the conditions
. We
of the proposition. Then, Proposition 1 implies
consider the case a ∈ Bσ 2 ( i ) and qk ∈ Bσ 3 ( i ) . We
that Con ( Bi ; f | Bi ) is a product of chains and
have b ∈ Bσ (i ) or b ∈ Bσ 3 ( i ) ; so, (a, k ) ∈ θ 3 for
Con( Bi ∪ B j ; f |Bi ∪ B j ) is a linear sum of a product
or {a, qk } ⊆ Bσ 3 ( i )
(a, qk ) ∈ θ 3 ; so, (a, b) ∈ θ3 ⊆ θ1 ∨ θ3
28
C. Ratanaprasert and S. Thiranantanakorn
Silpakorn U Science & Tech J Vol.5(1), 2011
of chains P with a one-element chain 1 for all
and (a, b) ∉ mi ; so {a, b} ⊄ Bi ∪ Bσ ( i ) . We may
i, j ∈ {1, 2,3} .
assume that a ∈ Bi , b ∈ Bσ 2 ( i ) . Let
If θ = D A , clearly θ Bi = D Bi
for all
If
i ∈ {1, 2,3} . Let θ ∈ ConA \{D A , A × A} and assume that θ ≠
3
θ i =1
Bi
( x, y ) ∈ mi .
. Then θ Bi ∪ B j ∪ θ Bk ⊆ θ
θ
or {x, y} ⊆ Bσ 2 ( i ) then
{x, y} ⊄ Bi ∪ Bσ (i )
If
and
{x, y} ⊄ Bσ 2 (i ) , we may assume that x ∈ Bi ∪ Bσ (i )
for all i ∈ {1, 2,3}. If ( x, y ) ∈ θ then {x, y} ⊄ Bi 3
{x,y} ⊆ Bi ∪ Bσ (i )
x, y ∈ A .
and
y ∈ Bσ 2 (i ) ;
( x, a ), (b, y ) ∈ mi ⊆ θ
so,
; so, there are
implies ( x, y ) ∈ θ . So, θ = A × A . Finally, let
1 ≤ i ≠ j ≤ 3 such that x ∈ Bi and y ∈ B j ; hence,
θ ∈ ConA which θ ⊄ mi for all i ∈ {1,2,3} .
( x, y ) ∈ θ Bi ∪ B j ∪ θ Bk where
{i, j , k} = {1, 2,3} .
There are (a, b) ∈ θ , (c, d ) ∈ θ , ( p, q ) ∈ θ ,
. or θ = θ Bi ∪ B j ∪ θ Bk where
(a, b) ∉ mi , (c, d ) ∉ mσ (i ) , ( p, q ) ∉ mσ 2 (i ) for
for all i ∈ {1, 2,3} since θ ≠
i =1
Therefore, θ =
3
θ i =1
Bi
Bi
{i, j , k} = {1, 2,3} .
Next,
θ = θ ∪θB
k
i ∈ {1,2,3} . Hence, a ∈ Bi ∪ Bσ (i ) , b ∈ Bσ 2 (i ) ,
β : (θ , φ ) θ ∨ φ
let
φ = φ ∪ D B ∪B
and
i
j
c ∈ Bσ (i ) ∪ Bσ 2 (i ) , d ∈ Bi , p ∈ Bσ 2 (i ) ∪ Bi and
where for
q ∈ Bσ (i ) . If a ∈ Bi , the cyclically of f and
all
( a, b) ∈ θ
θ ∈ Con( Bi ∪ B j ; f |B ∪ B ) and φ ∈ Con( Bk ; f | B ) . i
k
j
θt ∈ Con( Bi ∪ B j ; f |B ∪ B )
implies that ( x, y ) ∈ θ
for all
and
x, y ∈ Bi ∪ Bσ 2 (i ) . But ( p, q ) ∈ θ , we have either
φt ∈ Con( Bk ; f | B ) then θ t ∪ φt = θ t ∨ φt for
( s, t ) ∈ θ for all s, t ∈ Bσ (i ) ∪ Bσ 2 (i ) or ( s, t ) ∈ θ
t ∈ {1,2} . Thus, β is an order-embedding since
for all s, t ∈ Bi ∪ Bσ ( i ) . In any cases, ( s, t ) ∈ θ for
If
i
j
k
(θ1 , φ1 ) ⊆ (θ 2 , φ2 ) ⇔ θ1 ⊆ θ 2 and φ1 ⊆ φ2 ⇔ θ1 ∪ φ1 ⊆ θ 2 ∪ φ2
For each
⇔ θ1 ∨ φ1 ⊆ θ 2 ∨ φ2
i ∈ {1,2,3} , let Ci
⇔
β (θ1 , φ1 ) ⊆ β (θ 2 , φ2 )
all s, t ∈ A . Hence, θ = A × A . We can also prove
be a
that θ = A × A similarly if a ∈ Bσ (i ) .
sublattice of ConA which is isomorphic to
Con( Bi ∪ Bσ (i ) ; f | Bi ∪ Bσ ( i ) ) × Con( Bσ 2 (i ) ; f | B 2 )
Clearly, mi ∨ mσ ( i ) = A × A , for all
where {i, j , k} = {1, 2,3} and let mi be the greatest
i ∈ {1,2,3} . Let m be the greatest element of
element of Ci . We will show that m1 , m2 and m3
the sublattice C :=
are the only co-atoms of ConA . First, mi ≠ A × A
the greatest lower bound of {m1 , m2 , m3 } . So,
for i ∈ {1,2,3} since ( x, y ) ∉ mi for all i i and all
{m, m1 , m2 , m3 , A × A} is a sublattice of ConA
x ∈ Bi and y ∈ Bσ 2 (i ) . Secondly, let i ∈ {1,2,3}
which is isomorphic to M 3 . Therefore, ConA is
and mi ⊂ θ ⊆ A × A . Then, there exist (a, b) ∈ θ
not distributive.
σ (i )
3
C i =1
29
i
. It is clear that m is
Silpakorn U Science & Tech J Vol.5(1), 2011
All Congruence Modular Symmetric and Near-Symmetric Algebras
Note that : if θ , φ ∈ ConA with φ ⊆ θ
(i) L contains exactly three co-atoms m1 , m2
then θ , φ ∈ Ci for some i ∈ {1,2,3} . We will
and m3 where ↓mi satisfies Condition (i) of
now show that ConA has no sublattice which is
Corollary 1 for all i ∈ {1,2,3} , and
isomorphic to N 5 . Let θ , φ , ϕ ∈ ConA such that
φ ⊆ θ , ϕ | θ and ϕ | φ . Then, θ , φ ∈ Ci for
sublattice of L where m is the greatest element of
some 1 ≤ i ≤ 3 . If ϕ ∈ Ci , the distributivity of Ci
↓m .
(ii) The set {m, m1 , m2 , m3 ,1} forms a M 3 − 3
i =1
i
implies that φ ∧ ϕ ≠ θ ∧ ϕ and φ ∨ ϕ ≠ θ ∨ ϕ .
If ϕ ∈ C j for some 1 ≤ j ≠ i ≤ 3 then ϕ ∈ C
implies that φ ∧ ϕ ≠ θ ∧ ϕ and φ ∨ ϕ ≠ θ ∨ ϕ ;
Theorem 1 The followings are equivalence for a
and if ϕ ∈ C j \ C then θ ∈ Ci \ C and φ ∈ C
symmetric algebra A .
imply that θ ∨ ϕ = A × A and φ ∨ ϕ = m j ;
(i) A is congruence modular,
(ii) Conditions (i), (ii) or (iii) of Proposition
so,
φ ∨ϕ ⊂θ ∨ϕ ;
j
( j)
i
∪ ϕB (i )
(i )
σ 2 (i )
and
3 are satisfied,
which also imply that
σ 2 ( j)
θ ∧ ϕ = (θ B ∪ Bσ ∪ θ B i
θ , φ ∈ Ci \ C
θ = θ B ∪Bσ ∪ θ B
impliesthat
ϕ = ϕ B ∪Bσ
and
σ 2 (i )
) ∩ (ϕ B j ∪ Bσ ( j ) ∪ ϕ B 2 )
chain or a M 3 -head lattice.
σ ( j)
= φ ∧ ϕ . Therefore, ConA is modular.
(iii) ConA is either a product of chains , a
linear sum of a product of chains with one element
σ ( j)
⊂ (φBi ∪ Bσ ( i ) ∪ φB 2 ) ∩ (ϕ B j ∪ Bσ ( j ) ∪ ϕ B 2 ) σ (i )
We have the following characterizations.
¨
All lattices which are congruence lattices of
Corollary 1 If a symmetric algebra A is congruence-
modular near-symmetric algebras.
modular then there exist co-atoms m1 , m2 and m3
of ConA which satisfy the following conditions:
with l ( f ) = 1 which we will call A = ( A ; f ), a
(i) for each i ∈ {1,2,3} , ↓mi is one of the
Let f be a unary operation on a finite set A
near-symmetric algebra. The first proposition proves
lattices P , P ⊕ 1 or ( P ⊕ 1) × Q where P and
a characterizations of f .
Q are product of chains.
(ii) the set {m, m1 , m2 , m3 , A × A} forms a
Proposition 5. The followings are equivalent for a
M 3 − sublattice of ConA where m is the greatest element of
near-symmetric algebra A = ( A ; f ).
3
↓m . i =1
i
(i) l ( f ) = 1.
(ii)
There is a φ ≠ B ⊂ A such that
Definition 1 A lattice L with the greatest element
B ∩ Im f = φ and f
1 is said to be a M 3 -head lattice if
30
A\ B
is a permutation.
(iii) Im f ⊂ A and f
Im f
is a permutation.
C. Ratanaprasert and S. Thiranantanakorn
Silpakorn U Science & Tech J Vol.5(1), 2011
and Bi ∩ Im f for all blocks in A
(iv) Im f ⊂ A and B ∩ Im f is a one-
element set for all B ∈ A
permutation f
.
ker f Proof. Let l ( f ) = 1. Then Im f ⊂ A; so, there is
Im f
ker f
. Let the
be a product of disjoint cycles
a1 , a 2 ,..., a r for some r ≥ 2. If s ≥ 2, then | B1 | > 1
a φ ≠ B ⊂ A such that A = B ∪ Im f is a disjoint
and | B2 | > 1 ; hence, f ( B1 ) = {b1} ≠ {b2 } = f ( B2 );
union and f
so, there are u ∈ B1 and v ∈ B2 such that u , v ∉ Im f
Im f l ( f )
is a permutation.
If (ii) holds, then Im f ⊆ A \ B ⊂ A . Since
f
⊆ Im f ,
i ∈ {1, 2}; thus, ai and bi are in the same cycle for
we have A \ B ⊆ Im f . So, A \ B = Im f . Hence,
i ∈ {1, 2}. Let C be the union of cycles containing
f
A\ B
Im f
is a permutation and Im f
and there are a1 ≠ a2 such that ai ∈ Bi ∩ Im f for
A\ B
{b1 , b2 }. Then the congruences D A ⊂ θ (C ) ⊂
is a permutation.
Assume that (iii) holds. Let B ∈ A
ker f Then f ( B ) = {c} for some c ∈ A. Since f Im f
θ (C ) ∪ θ (u, v), ker f , and θ (C ) ∨ ker f will
.
generate a sublattice which is isomorphic to N 5 , a
is a permutation, B ∩ Im f is singleton. From (iv), we have f
and together with
Im f
contradiction. So, s = 1 .
is a permutation;
implies that A
Im f ⊂ A imply that
l ( f ) = 1.
¨
contains exactly one non-
ker f singleton block B. Since the least congruence
θ ( x, y ), θ ( y, z ) and θ ( x, z ) will generate a M 3
We note that ConA = {D A , ∇ A } for all two-
If A is congruence-distributive, Part (i)
-sublattice if x, y, z are distinct in B; so, B = 2.
elements algebras A, so we will consider algebras
Hence, Im f = | A | − 1 .
whose cardinalities are more than two.
¨
Proposition 7. If A is a near-symmetric Proposition 6. Let A be a near-symmetric algebra
algebra with | A | ≥ 4 and Im f =| A | − 1, then
with | A | ≥ 3 .
ConA ≅ 2 × Con(Im f ; f ).
(i) If A is congruence-modular then A
Proof. Let B be the only block of A
ker f
contains only one block whose cardinality more than
and b ∈ B ∩ Im f . Then θ ∪ {(u , u )} ∈ ConA and
one.
whose
ker f B = 2. Then f (u ) = f (b) for all u ∈ B \ Im f
θ := θ ∪ {( x, y )
(ii) If A is congruence-distributive then
x, y ∈ [b]θ ∪ {u}} ∈ ConA for
all θ ∈ Con(Im f ; f ). So, the map g : (1, θ ) → θ
Im f =| A | − 1 . Proof. Let A be congruence-modular. Proposition
and g : (0, θ ) →
5 tells the existence of the blocks B1 , B2 ,..., Bs
embedding from 2 × Con(Im f ; f ) into ConA.
whose cardinalities are more than one for some s ≥ 1
31
θ ∪ {(u, u )} is an order-
Now, let θ ∈ ConA. If [u ]θ is singleton,
Silpakorn U Science & Tech J Vol.5(1), 2011
All Congruence Modular Symmetric and Near-Symmetric Algebras
ϕ = θ \{(u, u )} ∈ Con(Im f ; f ) and g (0, ϕ ) = θ ;
congruence-modular with | A | ≥ 4. Then
and, if [u ]θ is not singleton, f (u ) = f (b) for all
(i) Im f =| A | − 1 or Im f =| A | − 2,
b ∈ B ∩ Im f ; so, g (1, ϕ ) = θ where ϕ is the
(ii)
corresponding congruence to the partition of Im f
ConA ≅ 2 × Con(Im f ; f ), and
containing the block of f (u ). Therefore, g is an
order-isomorphism.
ConA ≅ M 3 × Con(Im f ; f ).
Note that if Im f =| A | − 1 ≥ 3, then the
Im f =| A | − 1,
if
(iii)
if
then
Im f =| A | − 2,
Proof. Suppose that
then
Im f ≤| A | − 3. There
results in Remark 2 implies that Con(Im f ; f ) is
are distinct a, b, c, d ∈ A
not distributive. So, if Im f =| A | − 1 ≥ 3 or f
= f (c) = f (d ). So, the congruence θ (a, b),
Im f
with f (a ) = f (b)
is an identity, the proof of Proposition 7 shows that
θ (a, b) ∪ θ (c, d ) and θ (a, c) ∪ θ (b, d ) will
ConA is not distributive.
generate a N 5 − sublattice, a contradiction. But,
Im f ≤| A | − 1 implies
Im f =| A | − 2. One can see that (ii) follows from
Theorem 2. The followings are equivalent for a
Proposition 7. We assume that Im f =| A | − 2.
near-symmetric algebra A whose | A | ≥ 4 .
(i) A is congruence-distributive.
(ii)
Then, A
(iii) Im f =| A | − 1 and f
contains only one non-singleton
B = 2 then Im f =| A | − 1, a
contradiction. If B ≥ 4, , the proof of Proposition Im f
8 (i) implies a contradiction. Hence, B = 3.
is one of (i)
or (ii) of Proposition 1.
ker f
block B. If
Im f =| A | − 1 and (Im f ; f ) is
congruence-distributive.
Im f =| A | − 1 or
(iv) ConA is either 2 × P or 2 × ( P ⊕ 1)
Let B = {a, b, c}. Then f (a ) = f (b) = f (c)
and
B ∩ Im f = 1,
because
we
may
where P is a product of chains.
assume that c ∈ B ∩ Im f .
Proof. (i ) ⇒ (ii ) is clear from Proposition 7 and
M 3 = {D B , θ (a, b), θ (b, c), θ (a, c)θ ( B)} forms
(Im f ; f ) is congruence-distributive. (ii ) ⇒ (iii )
a M 3 − sublattice of ConA. Note that for each
follows from Proposition 1 and the argument
φ ∈ Con( B; f
after Proposition 7. (iii ) ⇒ (iv) is clear from
the relations φ = φ ∪ {( x, x) x ∈ Im f } and
Proposition 1, Proposition 4 and Proposition 7.
θ = θ ∪ {( x, x) x ∈ B} are in ConA. Now, let
Finally, (iv) ⇒ (i ) because the lattices 2 × P and
define
2 × ( P ⊕ 1) are distributive if P is a product of
β : ConA → M 3 × Con(Im f ; f ), by a (φ ,θ )
chains.
¨
= φ ∨θ
Proposition 8. Let A be a near-symmetric
and
32
B
)
and
Now, the set
θ ∈ Con(Im f ; f )
a : M 3 × Con(Im f ; f ) → ConA and for all (φ , θ ) ∈ M 3 × Con(Im f ; f )
β (θ ) (θ
B
,θ
Im f
) for all θ ∈ ConA,
C. Ratanaprasert and S. Thiranantanakorn
Silpakorn U Science & Tech J Vol.5(1), 2011
References Davey, B. A. and Priestley, H. A. (1990) Introduction to Lattices and Order. In Cambridge Mathematical Textbooks, New York. Denecke, K. and Wismath, S. L. (2002) Universal Algebra and Applications in Theoretical Computer Science. In Chapman & Hall, New York. Jakubikova, D. and Kosice (1982) On congruence relations of unary algebras I. Czechoslovak Mathematical Journal 32(107): 437 -459. Jakubikova, D. and Kosice (1983) On congruence relations of unary algebras I. Czechoslovak Mathematical Journal 33(108): 448 -466. McKenzie, R. and Hobby, D. (1998) The structure of finite algebras. Contemporary Mathematics vol. 76, Providence, Rhode Island. Ratanaprasert, C. and Denecke, K. (2008) Unary operations with long pre-periods. Discrete Mathematics 308: 4998 – 5005.
respectively. By routine work, one can prove that a β = idConA and β a = id M 3 ×Con (Im f ; f ) . Therefore, ConA ≅ M 3 × Con(Im f ; f ).
¨
Lemma 1. Let A be a near-symmetric algebra whose | A | ≥ 6 and either Im f =| A | − 1 or
Im f =| A | − 2. If f is not modular.
Im f
is an identity, ConA
¨
We have the following characterizations.
Theorem 3. The followings are equivalent for a near-symmetric algebra A whose | A | ≥ 4.
(i) A is congruence-modular.
(ii) (Im f ; f ) is congruence-modular and
either Im f =| A | − 1 or Im f = | A | − 2.
(iii) f
Im f
is one of (i) or (ii) or (iii) of
Proposition 6 and either Im f =| A | − 1 or Im f =
| A | − 2.
(iv) ConA is the lattice 2 × P , 2 × ( P ⊕ 1) ,
2 × L , M 3 × P , M 3 × ( P ⊕ 1) or M 3 × L where P is a product of chains and L is a M 3 -head lattice.
¨
33
Research Article Studies on Electrochromism of Chemically Deposited Nickel Oxide Thin Films Julijana Velevska,1* Margareta Pecovska-Gjorgjevich,1 Metodija Najdoski2 and Nace Stojanov1 1
Institute of Physics, 2Institute of Chemistry, Faculty of Natural Sciences and Mathematics, St. Ciril and Methodius University, P.O. Box 162, Skopje 1001, Macedonia * Coresponding author. E-mail address: julev@pmf.ukim.mk Received April 5, 2011; Accepted May 25, 2011
Abstract
In this work, nickel oxide (NiOx) thin films were prepared from aqueous solution of nickel nitrate
and carbamide. Those films were deposited onto fluorine doped tin oxide (FTO) coated glass substrates. An electrochromic test device (ECTD) was constructed by using these films as working electrodes, and FTO coated glass as an opposite electrode in alkaline medium of 0.1 M NaOH aqueous solution. The obtained nickel oxide films exhibited anodic electrochromism, changing color from transparent to dark brown. Various techniques were used for characterization of the films as x-ray diffraction (XRD), visible spectroscopy and electrochemical methods. The dependence of the optical density on the charge density was examined and the coloration efficiency was calculated from the optical density change. The time response was also determined. Key Words: Chemical bath deposition; Electrochromism; Nickel oxide
Introduction
Optically active thin films coatings can alter
material to change color reversibly, when a voltage
their optical properties as a function of changes
is applied across it. Actually, this is potentially
in external conditions. The changeable optical
the most commercially useful form of chromism.
characteristics can be obtained by different physical
An electrochromic process is complicated, and
and chemical processes in a large number of
the electrochromism of the different materials is
materials. These substances named chromogenics,
strongly related to the method of preparation, i.e.
embrace both inorganic and organic materials. The
it is affected by structure, stoichiometry, binding
mechanism responsible for the reversible variation
conditions and water content in the films.
of the optical properties of films may basically be
divided in three different types: thermochromism,
substances can be divided into two classes: inorganic
photochromism and electrochromism.
oxides and organic materials. Another classification
Silpakorn U Science & Tech J 5 (1) : 34-42, 2011
Electrochromism is the unique property of a
The various types of electrochromic
J.Velevska et.al
Silpakorn U Science & Tech J Vol.5(1), 2011
is related to the potential where coloration process
nickel oxide films prepared by CBD technique from
occurs. Cathodically coloring materials posses a
aqueous solution of nickel nitrate and carbamide.
reduced colored state, i.e. they color at a negative potential, while anodically coloring materials are
Materials and methods
those with oxidized color state, i.e. they color when
a positive potential is applied (Cerc Korosec and
were prepared by CBD method onto fluorine doped
Bukovec, 2006).
tin oxide (FTO) coated glass substrates. The CBD
Currently, the electrochromic properties
technique is very simple and does not required special
of nickel oxide films are of great interest for
set – up. A 100 ml – laboratory beaker was used for
applications as anodic material in electrochromic
deposition on four substrates. An aqueous solution
devices due to its high efficiency, low cost, and
containing 1 g Ni(NO3)2 · 6H2O and 3 g carbamide
good stability. There are varieties of techniques
in 80 ml deionized water was placed in the beaker.
for preparation of nickel oxide films. Usually they
Then the substrates were vertically introduced into
are prepared by cathode electrodeposition (Ragan
the solution. The temperature was adjusted and kept
et al., 1998), sputtering (Kitao et al., 1995), spray
at 95 °C. After 30 min, the substrates with deposited
pyrolysis (Arakaki et al., 1995), evaporation (Nagai,
films were taken out, washed under strong flash
1993], chemical vapour deposition (Maruyama and
water, rinsed with deionized water and dried in air.
Arai, 1993), sol – gel technique (Surca et al., 1996)
The thickness of the obtained films, measured with
etc.
Dectak Stylus Profilometer, was 200 nm.
Recently, soft solution processing (SSP)
Nickel oxide films investigated in this work
The composition and crystal structure of the
has raised considerable attention, as it is one –
films were studied by x-ray diffraction (XRD) by
step, environment friendly, and low – energy
Cu Ka radiation at wavelength λ = 1.54·10-10 m,
consumption process among existing techniques
with a Siemens D – 500 diffractometer.
of preparing thin films (Yoshimura, 1998). As a
branch of SSP, chemical bath deposition (CBD)
behavior of the obtained films, an electrochromic
has been well developed to fabricate the large –
test device (ECTD) was designed. It consisted of
area semiconductor thin films in view of many
a transparent cuvette with a 0.1 M NaOH aqueous
advantages: it does not require sophisticated
solution electrolyte in which two electrodes were
equipment like vacuum system and other expensive
immersed. The working electrode represented
instruments; the starting chemicals are commonly
the nickel oxide thin film onto FTO coated glass,
available and cheap; various substrates including
whereas counter electrode was FTO coated glass.
insulators, semiconductors or metals can be used;
Figure 1 depicts the cross section of the so – designed
the preparation parameters are easily controlled,
ECTD.
(Mane and Lokhnade, 2000), (Cheng et al., 2003),
(Todorovski et al., 2006), (Ristova et al., 2002),
films were studied using Varian CARY 50 Scan
(Igwe et al., 2009). As in any chemical method,
UV – Visible spectrophotometer, in the wavelength
composition of solution and thermal conditions has a
range from 320 to 820 nm. The visible transmission
strong effect on film properties. This paper presents
spectra were taken in-situ (the NiOx film incorporated
investigations of electrochromic properties of thin
as a working electrode into the ECTD) in both, the
35
In order to examine the electrochromic
The optical properties of the nickel oxide
Silpakorn U Science & Tech J Vol.5(1), 2011
Studies on Electrochromism of Chemically Deposited
The value of m is 1/2 for the direct allowed transition, 3/2 for forbidden direct allowed transition and 2 for indirect allowed transition. When the straight portion of the graph of (ahν )
m
against
hν is extrapolated to a = 0 the intercept gives the transition band gaps.
The cyclic voltammetry measurements were
carried out using computer linked three-electrode arrangement with platinum (Pt) as counter electrode and silver-silver chloride (Ag/AgCl) as a reference electrode in a potential range from –1 V to +1 V and back to –1 V versus the counter electrode. The
Figure 1 Cross-section of the ECTD: 1 – glass, 2 – FTO, 3 – NiOx film, 4 – electrolyte, 5 – light source, 6 – light detector.
voltage scan rate was 10 mV/s, and the film working area was 1 cm2.
The dependence of the transmittance at 670
nm laser light versus the applied cyclic voltage from –1 V to +1 V and back to –1 V was studied
completely bleached and colored states of the film.
with the following setup: light beam was directed
The blank probe data were taken as the working
to pass normally through the ECTD whereas a pin
electrode from the ECTD was replaced with FTO/
photodiode was mounted on the opposite side of
glass electrode, so that the transmission spectrum
the ECTD in order to receive the transmitted light
could be normalized to 100%. The coloration and
signal through ECTD. The transmittance was
the bleaching of the nickel oxide thin films were
determined by equation T = I I 0 , where I and I0
performed by application of a voltage +1 V and –1
are the photodiode current for a given voltage and
V respectively.
the photodiode current from the fully bleached film,
From the transmittance data and the film
respectively. The transmittance of the film’s fully
thickness (d = 200 nm) we determined the absorption
bleached state was fixed to 100%. The film working
coefficient from:
a=
area was 2.5 cm2.
1 1 ln d T
The most important parameter for the
characterization of the electrochromic film is the coloration efficiency CE (l) of a unit area at a
The optical gap E g of the film in the bleached
given wavelength. The coloration efficiency
and colored states was evaluated from the standard expression (Liu et al., 2005):
of the NiO x was calculated from the optical
a ⋅ h ⋅ν ∝ (h ⋅ν − E g ) m
670 nm, and the charge density D Q/S during
density change DOD = log Tb/Tc at wavelength
coloration after films were fully bleached:
where hν is the energy of the incident
photon, E g is the optical energy gap and m is the
CE ( l ) =
number which characterizes the optical processes.
36
S log (Tb Tc ) DQ
(1)
J.Velevska et.al
Silpakorn U Science & Tech J Vol.5(1), 2011
Where Tb and Tc are the transmittance (at
technique revealed anodic electrochromic behavior
wavelength 670 nm) of the fully bleached and
changing color from transparent to dark brown
colored states respectively, DQ is the ejected charge,
(Cerc Korosec and Bukovec, 2006). They could be
determined by the applied current through the test
repeatedly colored and bleached with alternative
device and the time of its application, and S is the
application of a positive and negative voltage
active area of the electrochromic device.
respectively, versus a counter electrode.
Finally, the responsiveness of the films was
The XRD pattern of the as deposited film
examined in situ at wavelength 550 nm (white light),
onto FTO coated glass substrate is presented in
as a transmittance change due to abrupt voltage
Figure 2. The seven most distinct detected peaks
change between –2 V and +2 V. The transmission at
were found to originate from the FTO substrate.
550 nm white light through the ECTD was registered
The other three detectible peaks at: 2.31·10-10 m,
with a pin photodiode, which served as a transmitted
2.096·10-10 m, and 1.474·10-10 m, were found to
light-measuring unit.
correspond to the nickel oxide. The conversion from D values to 2θ for these three peaks is given
Results and discussion
in Table 1. Diffraction peak at about 2θ = 38.940 is
value reported for β - phase Ni(OH)2 [Ezema et
The prepared nickel oxide films by CBD
d-spacing (Å) Figure 2 XRD pattern of a nickel oxide film onto FTO coated glass substrate.
37
Silpakorn U Science & Tech J Vol.5(1), 2011
Studies on Electrochromism of Chemically Deposited
Table 1 NiO peak identification and d spacing to
2θ conversio Peaks
d·1010 (m)
Figure 4 shows the in situ optical spectra of
the bleached and colored states of NiOx films in the wavelength range from 320 to 820 nm. These
2q (degrees)
data were taken relative to the FTO coated glass
1
2.31
38.94
substrate. As can be seen in this figure, there is a
2
2.096
43.11
significant difference in the transmittance between
3
1.474
62.98
the colored and bleached states of the films.
Figure 5 shows the plot of (ahν ) versus the 2
photon energy (hν ) for both, bleached and colored
states. A band gap is obtained from the intersection
al., 2008]. The diffraction peaks arising from NiO
of the fitted straight line and the abscissa. The
appear at about 2θ = 43.110 and 62.980 are values
similar energy value of 3.26 eV and 3.21 eV for
reported for cubic NiO [Jiao et al., 2003]. These
bleached and colored states of the films respectively
indicate coexistence of β - Ni(OH)2 and NiO phases.
was extracted. Comparing the results to the literature
Figure 3 presents photographs of NiOx films
data (3.2 - 3.6 eV), (Wang et al., 2005; Subramanian
in a) bleached and b) colored states, which were
et al., 2008; Sonavane et al., 2010),we find similar
made ex situ. The films were first removed from the
values with the same type of transition.
ECTD after bleaching (coloring), rinsed in distilled
water, dried, and then photographed with a digital
The current change in Figure 6 and the
transmittance change at λ= 670 nm in Figure 7 are
camera. One could notice significant difference
presented as a function of the voltage applied across
between the colors of the films in both states.
the full device (Rubin et al., 1998). The ECTD was cycled over range ±1 V. The voltage sweep rate of cycling was 10 mV/s. The transmittance at
λ = 670 nm of the ECTD in a fully bleached state was taken to be 100%. The graph presents the histogram of I – V, which is due to the electrochromic behavior, whereas the transmittance (at λ= 670 nm) changed from 100% (normalized transmittance for the bleached state) to 40% corresponding to the colored state.
Figure 8 shows the graphical presentation
of the optical density change (DOD) as a function of the charge density Q/S. Since DOD changes linearly with Q/S, the coloration efficiency (CE) can be determined from the slope of the straight line in Fig. 8. The coloration efficiency calculated from Equation 1 was found to be 40.6 cm2/C, which is good value compared with those obtained for the
Figure 3 Photographs of a NiOx films in colored
NiOx films deposited by other techniques (Ferreira
and bleached states.
38
J.Velevska et.al
Silpakorn U Science & Tech J Vol.5(1), 2011
Figure 4 Optical transmission of the NiOx films in the bleached and colored states in the visible part of the spectrum. The data were taken relative to the FTO coated glass substrate.
Figure 5 Plots of (ahν ) as a function of photon 2
Figure 6 A cycling voltammogram of the NiOx
energy (hν ) for NiOx films in bleached
films recorded from –1 V to +1 V and back to –1V in a three-electrode arrangement
and colored states.
with Pt as counter electrode and an Ag/ AgCl as a reference electrode in a 0.1 M NaOH aqueous solution.
39
Silpakorn U Science & Tech J Vol.5(1), 2011
Studies on Electrochromism of Chemically Deposited
et al., 1996; Lechner and Thomas, 1998; Kadam and Patil, 2001; Ristova et al., 2002; Velevska and Ristova, 2002; Park et al., 2004).
In order to examine the time response
(coloration and bleaching times), the transmittance for white light (λ= 550 nm) was measured in situ trough the ECTD (Figure 9). The applied potential was switched between –2 V and +2 V. The transmittance of the ECTD at 550 nm in fully bleached state was normalized to 100%. The coloration and bleaching times, defined as the time to reach 2/3 of the final change in the transmittance
Figure 7 Presentation of the transmittance change
(Cinnsealach et al., 1998), were found to be about
at 670 nm of the NiOx films recorded
4 and 2.6 s, respectively. It means that NiOx films
from –1 V to +1 V and back to –1 V
prepared by CBD technique can reach their bleached
against FTO electrode.
state more rapidly than colored state. It could be explained with the lower conductivity of Ni(OH)2 (bleached state) than that of the NiOOH (colored state) (Ahn et al., 2002). Conclusions
Nickel oxide thin films were deposited
onto FTO coated glass substrates from aqueous solution of nickel nitrate and carbamide using CBD technique. The technique is simple, economic, and offers a possibility for large area depositions. The obtained films exhibited electrochromic behavior:
Figure 8 The dependence of the optical density on
they were colored at a positive potential, changing
charge density.
color from transparent to dark brown. The optical and the electrochromic properties of this material are convenient for visible light intensity modulation applications. Variations in the transmittance for 670 nm laser light due to the charge insertion/extraction of about 60% were achieved. Obtained nickel oxide films show fast response time (about 4 s for coloring and 2.6 s for bleaching). Finally, the coloration efficiency (at 670 nm) was found to be 40.6 m2/C,
Figure 9 The time response of the transmittance
which is good value compared with those obtained
recorded at 550 nm. The applied potential
for the NiOx films deposited by other techniques.
is switched between –2 V and +2 V.
40
J.Velevska et.al
Silpakorn U Science & Tech J Vol.5(1), 2011
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