Cmjv02i01p0093 by Journal of Computer and Mathematical Sciences

J. Comp. & Math. Sci. Vol.2 (1), 93-97 (2011)

Unitary Quadnary System RAMESH CHANDRA1 and S. K. GANTAYAT 2 3/3 Girls Normal School1 Campus, Faizabad, U.P. India email : rameshchandra51@gmail.com Department of Mathematics, Berhampur City College2, Berhampur, Ganjam, Orissa, India email: dr.susilgantayat@gmail.com ABSTRACT It is shown that a quadnary system can have none, or 3n, n = 0,1,2,….., or infinite units. Key words: Quadnary System, Unit, Unitary Quadnary System, Regualr Conjugate Regular Quadnary System, Singular Quadnary System. AMS Subject Classification : 08A99

INTRODUCTION Let X be a non-empty set and ω3 be a quadnary operation on X.Let a,b,c,d ∈X. For simplicity, we write ω3 (a,b,c,d) as (a,b,c,d). The operation ω3 is called “associative” if ∀ a,b,c,d,e,f,g ∈ X, (2.1)

[(a,b,c,d),e,f,g] = [a,(b,c,d,e),f,g] = [a,b,(c,d,e,f),g] = [a,b,c,(d,e,f,g)].

The system (X, ω3), where ω3 is associative, is called a quadnary system.1 In what follows, we define a quadnary system by Q.Thus Q = (X, ω3); and to x ∈ X,

for

simplicity, we write x ∈ Q.. A quadnary system Q is said to be unitary iff ∃ e ∈ Q such that, (2.2) (i) ∀ a ∈ Q, ( e,a,e,e) = a (2.3) (ii) ∀ a ∈ Q,∃ ā ∈ Q such that (a,e,e, ā ) = (ā,e,e,a) = e.

We denote a quandary system Q,with unit e , as [Q,e] or Q(e) or Qe In a unitary quandary system [Q,e], e is called the ‘Unit’. And ā in (2.3) is called ‘inverse” of a in the system. Example : Let Q = {a,b}, where ω4 on Q is defined as follows. Let (x,y,z,u) denote a quadruplet (x,y) (z,u) (a,a) (a,b) (b,a) (b,b)

(a,a) a b b a

(a,b)

(b,a)

(b,b)

b a a a

a b a a

b a a b

(2.5) Here, ‘a’ is a unit in Q and ā = b, bɸ = a. Lemma 1: algebra (Q, e)

a unitary

Journal of Computer and Mathematical Sciences Vol. 2, Issue 1, 28 February, 2011 Pages (1-169)

quandary

Ramesh Chandra, et al., J. Comp. & Math. Sci. Vol.2 (1), 93-97 (2011)

(i) (e,e,e,e) = e (ii) ∀ a ∈ Q (a,e,e,e)=(e,e,a,e) = (e,e,e,a) = a (iii) ∀ a ∈ Q, (āɸ) = a (iv) ∀ a ∈ Q, (a,e, ā,e ) = (a, ā,e,e)= (ā,e,a,e) =( ā,a,e,e) = e. (v) ∀ a , ā is unique.

Let Q be the quandary system. An element a ∈Q is called regular (2.6)iff ∃ b ∈Q such that, ( a,(b,a,b,a),b,a) = a .

Proof : (i) Putting a = e we get, (e,e,e,e) = e (ii) (a,e,e,e) = [(e,a,e,e),e,e,e] = [e,a,e,(e,e,e,e)] = [e,a,e,e] = a.

If a and b are two elements of Q such that, (a, (b,a,b,a), b, a) = a, and (b, (a,b,a,b), a, b) = b, (2.7) then a, b are said to be regular conjugates to one another. Another element a ∈ Q can have more than one regular conjugates.

Similarly,(e,e,a,e) = [e,e(e,a,e,e),e] = [e,a,e,(a,e,e,e)] = (e,e,e,a) = [e,e,e,(e,a,e,e)] = (e,a,e,e) = a.

A quadnary system Q is called a regular quandary system if every element in Q (2.8) is regular and has a regular conjugate.

We have (ā,e,e, āɸ) = e ⇒ [a,e,e, (ā,e,e, āɸ)] = (a,e,e,e) ⇔ [(a,e,e, ā),e,e, āɸ] = a ⇔ (e,e,e, āɸ) = a. ⇒ āɸ = a. (iii) We have (a,e, ā,e) = [a,e,(e, ā,e,e),e] = [a,e,e(ā,e,e,e)] = [a,e,e, ā]. Similarly we have (a, ā,e,e) = [(e,a,e,e), ā,e,e] = [e,(a,e,e, ā),e,e] = [e,e,e,e] = e. Following similarities, we can show that (ā,e,a,e) = (ā,a,e,e,) = e. (iv) Suppose there is b also such that (a,e,e,b) = e. then [a,e,e,b] = [a,e,e, ā] ⇒ [ ā,a,e(a,e,e,b)]= [ā,e,e,(a,e,e, ā)] ⇔ [ (ā,e,e,a),e,e,b] = [(ā,e,e,a),e,e, ā] ⇔ [e,e,e,b] = [e,e,e, ā] ⇒ b = a. Thus ā is unique.

A quadnary system Q is called a singular quandary system iff each element in it (2.9) has a unique regular conjugate. Lemma 2 : Let [Q,e] be a unitary system. Then ā is the regular conjugate of a. Proof :

We have [a,( ā, a, ā, a), ā, a] = [(e, a, e, e),( ā, a, ā, a), ā, a] = [e,(a, e, e, ā),(a, ā, a, ā),a] = [e, e,(a, ā, a, ā),a] = [e, e,{(e, a, e, e), ā, a, ā},a] = [e, e,{e,(a, e, e, ā),a, ā},a] = [e, e,(e, e, a, ā),a] = [e, e, e, a] = a. Thus ā is the regular conjugate of a. Corollary : Every [Q, e] is singular quadnary system. Proof : Let a∈ Q. Then ā is the regular conjugate of a. as shown earlier ā is unique.Thus [Q,e] is singular quadnary system.

Journal of Computer and Mathematical Sciences Vol. 2, Issue 1, 28 February, 2011 Pages (1-169)

Ramesh Chandra, et al., J. Comp. & Math. Sci. Vol.2 (1), 93-97 (2011)

Lemma 3: A singular quadnary system is unitary if, ∃ e ∈ Q such that ∀ a ∈ Q (e,a,e,e) = a

Proof : Let xɸ, yɸ, zɸ denote the inverses of respectively x,y,z in [Q,e]. Then

Proof : We show that e is a unit. Putting a = e, we get (e,e,e,e) = e. Further [e,(e,e,e,e),e,e] = e. Thus e is a conjugate of e. Further let a ∈ Q, ā denote the conjugate of a, we show (a,e,e, ā) = e. Put ( a,e,e, ā) = ao . Then , ao= [ {a,( ā,a, ā,a), ā,a},e,e, ā] =[a,{(e,e, ā,e),a, ā,a}, ā,a},e,e, ā] =[(a,e,e, ā),{(e,a, ā,a), ā,a,e},e, ā] =[ao,e,{(a, ā,a, ā),a,e,e}, ā] =[ao,e, {(a,(e,e, ā,e),a, ā),a,e,e}, ā] =[ao,e,{ao,e,a, ā},(a,e,e, ā)] =[ao,e,{(ao,e,a, ā},ao] =[ao,e,{ao,e,a,(e,e, ā,e)},ao] =[ao,e,{ao,e,(a,e,e, ā),e},ao] =[ao,e,{ao,e,ao,e},ao] =[ao,{e,ao,e,ao},e,ao]

(i) (x,a,b,c)= [x,a1,b1,c1]⇒[xɸ , e,e, (x,a,b,c)]

Thus e has regular conjugate ⇒ ao is the regular conjugate of e. But e is the regular conjugate of e and Q is singular⇒ the conjugate of e is unique ⇒ ao = e. Hence, (a,e,e, ā) = e. Thus, e is a valid unit and [Q,e] is a unitary quadnary system. Lemma 4: A Unitary Quadnary system is cancellative. For x,y,z ,a,b,c, a1,b1,c1 ∈ [Q,e] (i) (x,a,b,c) = (x,a1,b1,c1) ⇒ (e,a,b,c) = (e,a1,b1,c1) (ii) (x,a,b,y) = (x,a1,b1,y) ⇒ (e,a,b,e) = (e,a1,b1,e) (iii) (x,y,z,a)= (x,y,z,b) ⇒ a = b.

= [xɸ,e,e,(x,a1,b1,c1)] ⇔[ (xɸ,e,e,x),a,b,c] = [(xɸ,e,e,x),a1,b1,c1] ⇔[e,a,b,c] = [e,a1,b1,c1] (ii) (x,a,b,y) = (x,a1,b1,y) ⇒ [{ xɸ,e,e,(x,a,b,y)},e,e,yɸ] = [{ xɸ,e,e,(x,a1,b1,y)},e,e,yɸ] ⇔ [( xɸ,e,e,x),a,b,(y,e,e,yɸ)] = [( xɸ,e,e,x),a1,b1,(y,e,e,yɸ)] ⇔ [e,a,b,e] = [e,a1,b1,e]. (iii) (x,y,z,a) = (x,y,z,b) ⇒ [ xɸ,e,e,(x,y,z,a)] = [ xɸ,e,e,(x,y,z,b)] ⇔[( xɸ,e,e,x),y,z,a] = [(xɸ,e,e,x),y,z,b] ⇔[e,y,z,a] = [e,y,z,b] ⇔[y,z,e,a] = [y,z,e,b] ⇔[yɸ ,e,e(y,z,e,a)] = [yɸ,e,e,(y,z,e,b)] ⇔ [(yɸ,e,e,y),z,e,a] = [(yɸ,e,e,y),z,e,b] ⇔ [e,z,e,a] = [e,z,e,b] ⇔ [z,e,e,a] = [z,e,e,b] ⇔ [e,e,e,a] = [e,e,e,b] by (i) ⇒ a = b. Lemma 5 : Let [Q, e] is Unitary quadnary system. Then an element x ∈ Q is another unit iff, (x, a, x, x) = x Proof : [Q,e] is unitary system ⇒ Q is singular quadnary system. Thus Q is singular, and x ∈ Q such that, ∀ a ∈ Q (x,a,x,x) = a Hence x is a unit.

Journal of Computer and Mathematical Sciences Vol. 2, Issue 1, 28 February, 2011 Pages (1-169)

Ramesh Chandra, et al., J. Comp. & Math. Sci. Vol.2 (1), 93-97 (2011)

Corollary : Let Q be a singular quadnary system. Then Q has n units iff, (x,a,x,x) = a , ∀ a ∈ Q has n solutions. Here we give an example of a quadnary system having three units

(a,a)

Example : Let Q = [a,b,c,d]. Let ω4 on Q be defined as follows. Let (x,y,z,u) be a general quadruplet. We define:

(a,a) (a,b)

(a,c) (a,d) (b,a) (b,b)

(b,c)

(b,d)

(c,a)

(c,b)

(c,c) (c,d)

(d,a) (d,b)

(d,c) (d,d)

(a,b)

(a,c)

(a,d)

(b,a)

(b,b)

(b,c)

(b,d)

(c,a)

(c,b)

(c,c)

(c,d)

(d,a)

(d,b)

(d,c)

(d,d)

Here, (a,c,b,d) = a . It may be easily verified that a Quadnary system may have more than one unit. The question then arises as to how many? The following theorem shows that the number of units shall be 3n, where n = 0,1,2…… The number of units in a finite unitary quadnary system can be 0,1,3,32, ….3n . Where n is a non-negative integer. Proof : Let [Q, e] be a unitary quadnary system. We define, a, b ∈ Q a * b = (a, e, e, b) We show [Q, * ] is a group.

G0 : G1 :

G2:

Clearly ‘ *’ is closed on Q Next, we show [Q, * ] is associative. Let a, b, c ∈ Q. Then (a * b) * c = [(a, e, e, b),e, e, c] =[a, e, e,(b, e, e, c)] = a * (b * c) [Q, *] has an identity element ‘e’. Let a ∈ Q. Then a * e = (a, e, e, e) = a

G3 : ∀ a ∈ Q, ∃ aɸ , the regular conjugate of a in [Q, e ]. Then a * ā = (a, e, e, ā) = e . Thus [Q , * ] is a group. Let U denotes all the possible units of Q

Journal of Computer and Mathematical Sciences Vol. 2, Issue 1, 28 February, 2011 Pages (1-169)

Ramesh Chandra, et al., J. Comp. & Math. Sci. Vol.2 (1), 93-97 (2011)

Claim: U is a commutative subgroup of [Q, *] First we show that ‘*’ commutative over U. Let e1, e2 be two units of Q, that is, e1, e2 ∈ U, then e1 * e2 = (e1, e, e,e2) = [(e2, e1, e2,e2),e,e,e2] = [e2,(e1,e2,e2,e2),e, e] = [ e2,e1,e,e] = [ e2,e,e,e1] = e2 * e1 Next we show that U is closed under ‘*’. For this, we need to show that e1,e2 ∈ U, e1 * e2 also ∈ U. For this we need to show that ∀a ∈ Q [e1 * e2 , a , e1 * e2 , e1 * e2] = a As e1,e,e2 are all units, they are commutable, so that [e1 * e2, a, e1 * e2, e1 * e2] =[(e1,e,e,e2), a, (e1,e,e,e2), (e1,e,e,e2)] can be reshuffled to [e1 * e2 , a , e1 * e2 , e1 * e2] [(e, e1,e1,e1), a1, (e,e2,e2,e2), (e,e,e,e)] = [e, a, e, e] = a. Thus e1 * e2 is a unit. Hence [U, * ] is a commutative subgroup of [Q, * ] .

Now , let us consider a finite set X. let Ro, R1, R2 denote the residue classes modulo 3. {Ro, R1, R2}] Let [ Fx . f . X Then [Fx, +] is a commutative group, fo defined by fo (x) = Ro. By suitably choosing x, we can make Fx have the same cardinality as U . f1 e1 then If f1 ∈ Fx f0 + f1 + f2 ( e 1 * e 1) * e 1 = [{(e1,e,e,e1),e, e,(e1,e,e,e1)},e,e,e1] The right hand side can be reshuffled to = [(e1,e1,e1,e),e, e,(e, e, e, e)] = [e, e, e, e] = e. It is easy to verify f * f * f = fo. Thus, we can have Fx is isomorphic to U Hence, U = F(x)  = 3 x  Thus the number of units in a unitary quadnary system will be 30 ,31 ,32 ,33, ……….3n, or n 1,3,9,27,……3 . REFERENCES 1. H. P. Sankappannavar and S. Burris : “A Course in Universal Algebra”, Springer & Verlag, 19812.

Journal of Computer and Mathematical Sciences Vol. 2, Issue 1, 28 February, 2011 Pages (1-169)