Semigroups

Pacific Journal of Mathematics, 1970

2011

Pacific Journal of Mathematics, 1975

Semigroup Forum, 2015

Double semigroups have two associative operations •, • related by the interchange relation: (a • b) • (c • d) ≡ (a • c) • (b • d). Kock [13] (2007) discovered a commutativity property in degree 16 for double semigroups: associativity and the interchange relation combine to produce permutations of elements. We show that such properties can be expressed in terms of cycles in directed graphs with edges labelled by permutations. We use computer algebra to show that 9 is the lowest degree for which commutativity occurs, and we give self-contained proofs of the commutativity properties in degree 9.

In this paper, the notion of an ordered Γ-semigroup is introduced and some examples are given. Further the terms commutative ordered Γ-semigroup, quasi commutative ordered Γ-semigroup, normal ordered Γsemigroup, left pseudo commutative ordered Γsemigroup, right pseudo commutative ordered Γsemigroup are introduced. It is proved that (1) if S is a commutative ordered Γ-semigroup then S is a quasi commutative ordered Γ-semigroup, (2) if S is a quasi commutative ordered Γ-semigroup then S is a normal ordered Γ-semigroup, (3) if S is a commutative ordered Γ-semigroup, then S is both a left pseudo commutative and a right pseudo commutative ordered Γ-semigroup. Further the terms; left identity, right identity, identity, left zero, right zero, zero of an ordered Γ-semigroup are introduced. It is proved that if a is a left identity and b is a right identity of an ordered Γ-semigroup S, then a = b. It is also proved that any ordered Γsemigroup S has at most one identity. It is proved that if a is a left zero and b is a right zero of an ordered Γsemigroup S, then a = b and it is also proved that any ordered Γ-semigroup S has at most one zero element. The terms; ordered Γ-subsemigroup, ordered Γsubsemigroup generated by a subset, α-idempotent, Γ-idempotent, strongly idempotent, midunit, r-element, regular element, left regular element, right regular element, completely regular element, (α, β)-inverse of an element, semisimple element and intra regular element in an ordered Γ-semigroup are introduced. Further the terms idempotent ordered Γ-semigroup and generalized commutative ordered Γ-semigroup are introduced. It is proved that every α-idempotent element of an ordered Γ-semigroup is regular. It is also proved that, in an ordered Γ-semigroup, a is a regular element if and only if a has an ( , )-inverse. It is proved that, (1) if a is a completely regular element of an ordered Γ-semigroup S, then a is both left regular and right regular, (2) if "a" is a completely regular element of an ordered Γ-semigroup S, then a is regular and semisimple, (3) if "a" is a left regular element of an ordered Γ-semigroup S, then a is semisimple, (4) if "a" is a right regular element of an ordered Γ-semigroup S, then a is semisimple, (5) if "a" is a regular element of an ordered Γsemigroup S, then a is semisimple and (6) if "a" is a intra regular element of an ordered Γsemigroup S, then a is semisimple. The term separative ordered Γ-semigroup is introduced and it is proved that, in a separative ordered Γ-semigroup S, for any x, y, a, b ∈ S, the statements (i) xΓa ≤ xΓb if and only if aΓx ≤ bΓx, (ii) xΓ xΓa ≤ xΓxΓb implies xΓa ≤ xΓb,(iii) xΓyΓa ≤ xΓyΓb implies yΓxΓa ≤ yΓxΓb hold.

2015

This paper concerned with basic concepts and some results on (idempotent) semigroup satisfying the identities of three variables. The motivation of taking three for the number of variables has come from the fact that many important identities on idempotent semigroups are written by three or fewer independent variables. We consider the semigroup satisfying the property abc = ac and prove that it is left semi-normal and right quasi-normal. Again an idempotent semigroup with an identity aba = ab and aba = ba (ab = a, ab = b) is always a semilattices and normal. An idempotent semigroup is normal if and only if it is both left quasi-normal and right quasi-normal. If a semigroup is rectangular then it is left and right semi-regular.

Journal of Algebra, 1998

International Journal of Mathematics Trends and Technology, 2016

There is a close relationship between automata and semigroups. Automata includes calculating machines, computers, telephone switch boards, elevators and so on. Algebraic automata theory make extensive use of semigroups. The aim of this paper is to summarize the results of semigroup theory which will be described through (semi)automata. For different types of (semi-)automata, we identified different structures of semigroups and their properties.

Portugaliae Mathematica

In this note we prove a selection of commutativity theorems for various classes of semigroups. For instance, if in a separative or completely regular semigroup S we have x p y p = y p x p and x q y q = y q x q for all x, y ∈ S where p and q are relatively prime, then S is commutative. In a separative or inverse semigroup S, if there exist three consecutive integers i such that (xy) i = x i y i for all x, y ∈ S, then S is commutative. Finally, if S is a separative or inverse semigroup satisfying (xy) 3 = x 3 y 3 for all x, y ∈ S, and if the cubing map x → x 3 is injective, then S is commutative.

Semigroup Forum, 2013

An algebraic structure is finitely related (has finite degree) if its term functions are determined by some finite set of finitary relations. We show that the following finite semigroups are finitely related: commutative semigroups, 3-nilpotent monoids, regular bands, semigroups with a single idempotent, and Clifford semigroups. Further we provide the first example of a semigroup that is not finitely related: the 6-element Brandt monoid.