ON LEFT ALMOST SEMIGROUPS, RIGHT ALMOST SEMIGROUPS AND GROUPOIDS CONTAINING COMMUTATIVE ORTHODOX SEMIGROUPS AND COMMUTATIVE GROUPS
Shah, Syed Aleem,, Nisar Ahmad, Syed, Aleem, Shah, Nisar, Kohat, Kust · INRIA a CCSD electronic archive server · 2020
In this article we have constructed some new patterns of finite left almost semigroups and right almost semigroups in such way that its idempotent along with some non idempotent(s) form commutative orthodox semigroups and commutative groups and also discussed new patterns of groupoids containing cyclic groups and orthodox semigroups. We also solved open problem put by N. Ahmad et al [11]. Preliminaries: In literature left almost semigroup abbreviated as LA-Semigroup is a structure S for all inputs a, b and c ϵ S the condition (ab)c = (cb)a holds [6]. Similarly a groupoid S is called right almost semigroup abbreviated as RA-Semigroup for all a, b and c ϵ S, a(bc) = c(ba) [6]. A groupoid S is called medial medial law or bisymmetry law is satisfied for all a, b, c and d ϵ S, (ab)(cd) = (ac)(bd) [6]. A groupoid S is called double displacement groupoid for all a, b, c and d ϵ S (ab)(cd) = (ad)(bc) [11]. A semigroup S is called complete regular semigroup for all a, b ϵ S the conditions aba = a, a 2 b = a and ba 2 = a all are satisfied. A semigroup S is called E-semigroup the subset of S (say S1) containing all of its idempotents also form semigroup w.r.t binary operation defined on set S. A semigroup S is called Orthodox semigroup S is E-semigroup as well as regular semigroup. A groupoid S is called locally associative for all a ϵ S condition a 2 a = aa 2 is satisfied [4]. If each element a ϵ S is idempotent then S is locally associative groupoid. LA-Semigroup S is called LA-Group there exists left identity and left inverse of each element exists. RA-Semigroup S is called RA-Group there exists right identity and right inverse of each element exists. A groupoid S is called unipotent groupoid S contains only one idempotent a such that a 2 = a and for all b in S, b 2 = a. Introduction: D. Mclean [1], presented concept of idempotent semigroups and explained the conditions for which idempotents of a semigroup also form semigroup. N. Kimura [2], explained the idea of idempotent semigroups in their research and explained details about left regular, right regular and regular semigroups. Yamada and Kimura [3], discussed about the left (right) normality of semigroup and proved that semigroup S is left (right) normal then S is left (right) regular. Clifford and Preston [4], elaborated the idea that if I is an ideal of a semigroup S, then I and S/I are regular (inverse) and only S is regular (inverse). Clifford and Preston also proved the result that R is some zero minimal right ideal of S then either rS = R for each r belongs to R\0 or else rS = 0 is satisfied with R = {0, r}. T.E. Hall [5], presented idea of such structures which are regular semigroups and their idempotents also form regular semigroup. T.E. Hall discussed the concept of commutativity, left normality, right normality and normality and achieved his results. Kazim and Naseeruddin [6], introduced the concept of left almost semigroup in 1971 during their Ph.D studies they proved that every commutative semigroup is LA-Semigroup as well as RA-Semigroup. Mushtaq and Yousaf [7], extended the work of Kazim and Naseeruddin in their research and developed the idea of LA-Semigroups which are locally associative. Q. Mushtaq [8], extended these results and introduced the concept that on what conditions left almost semigroup becomes commutative monoid and becomes group. N. Kehayopulu [9], used specific multiplication and ordering of elements and constructed semigroups which are completely regular. N.