Seira-Group

If we make a neutrosophic law * defined as: x1(t1,i1,f1) * x2(t2,i2,f2) we might need two classical laws # and <>, one that combines the elements x1#x2, and the other that combines their neutrosophic coordinates (t1,i1,f1)<>(t2,i2,f2). I have constructed this example: x1(t1, i1, f1) * x2(t2, i2, f2) = max{x1, x2}( min{t1, t2}, max{i1, i2}, max{f1, f2}). Using this example, we can construct a (t,i,f)-neutrosophic group/loop, and then a (t,i,f)-neutrosophic multiset group/loop. This is a case when # and <> work separately, But it might be possible to combine x1, x2, t1,i1,f1, t2,i2,f2 all together...

Seira-Group

Etymologically, (Gr.) σειρά [ seirá ] means set, string. Therefore, Seira-Group is a group whose unitary element is replaced by a set of two or more distinct elements. Definition of Seira-Group Let (G, E, *) be a nonempty set, where the law * is: - Well-Defined: if x, y ∊G, then x*y ∊G; - Associative: if x, y, z ∊ G, then (x*y)*z = x*(y*z); - The set E ⊊G (i.e. E ≠ G) has at least two distinct elements: E = {e1, e2, …, en}, for n ≥ 2.

Each element x ∊ G has at least one corresponding unit (neutral) element e ∊ E, i.e. x*e = e*x = x. And for each element ei ∊E there exist at least one element xi ∊G, such that xi*ei = ei*xi = xi, with i ∊ {1, 2, …, n}. {Therefore, the classical unit (neutral) element is replaced by a set of two or more distinct unit (neutral) elements.} - For each element x ∊ G there exist at least an inverse element x-1 ∊G such that x*x-1 = x -1*x ∊E. If the law * is commutative, then (G, E, *) is a Commutative

Seira-Group. Example 1 of Seira-Group Using the Cayley Table: G = {a, b, c}, E = {a, b}. * a b c a a a b b b b c c a c b The neutrals are: neut(a) = a, neut(b) = b, neut(c) = b. The inverse elements are: anti(a) = a, anti(b) = b, anti(c) = c. (G, *) is also a Neutrosophic Triplet Group, whose neutrosophic triplets are: <a, a, a>, <b, b, b>, <c, b, c>.

Theorem 1. A Seira-Group is also a Neutrosophic Triplet Group. Proof: Let (G, E, *) be a Seira-Group. The law * is well-defined, and associative. For each element x ∊ G, there exist at least a neutral element that we denote ex ∊ E ⊂ G and at least an inverse element that we denote x-1∊ G, therefore a neutrosophic triplet (x, ex, x-1) included in G. Therefore (G, E, *) is also a Neutrosophic Triplet Group. Theorem 2. If a Neutrosophic Triplet Group has at least two distinct neutral elements, then it is also a Seira-Group. Example 2 of non Seira-Group We change the law * to #. G = {a, b, c}, E = {a, b}. # a b c a a a a b b b c c a c a The unit/neutrals are: neut(a) = a, neut(b) = b, neut(c) = b. The inverse/anti elements are: anti(a) = a, anti(b) = b, anti(c) = none. Therefore (G, #) is not a Seira-Group (since “c” has no inverse), nor a Neutrosophic Triplets Group, because: <a, a, a>, <b, b, b>, are neutrosophic triplets, but