Structure of Regular Semigroups

R. H. Bruck's theorem [1] established the fact that any semigroup S can be embedded in a Simple Semigroup which posses an identity element. In this paper, we discuss some of the properties which shares with any semigroup which posses an identity element. Thus we establish the following results i. Any regular (inverse) semigroup can be embedded in a Simple regular (Inverse) semigroup with an identity element ii. There exist simple inverse (and hence, regular) semigroups with an identity element which have an arbitrary cardinal number of D – classes. These results are new extensions arising from [1].

Algebra Universalis, 1988

Semigroup Forum, 2002

We provide short and direct proofs for some classical theorems proved by Howie, Levi and McFadden concerning idempotent generated semigroups of transformations on a finite set.

Pacific Journal of Mathematics, 1975

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.

Journal of Algebra, 1998

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.

Proceedings of the Edinburgh Mathematical Society, 1979

A monoid in which every principal right ideal is projective is called a right PP monoid. Special classes of such monoids have been investigated in (2), (3), (4) and (8). There is a well-known internal characterisation of right PP monoids using the relation ℒ* which is defined as follows. On a semigroup S, (a,b) ∈ℒ* if and only if the elements a,b of S are related by Green's relation ℒ* in some oversemigroup of S. Then a monoid S is a right PP monoid if and only if each ℒ*-class of S contains an idempotent. The existence of an identity element is not relevant for the internal characterisation and in this paper we study some classes of semigroups whose idempotents commute and in which each ℒ*-class contains an idempotent. We call such a semigroup a right adequate semigroup since it contains a sufficient supply of suitable idempotents. Dually we may define the relation ℛ* on a semigroup and the notion of a left adequate semigroup. A semigroup which is both left and right adequate w...

Journal of Algebra, 2011

Let a be a non-invertible transformation of a finite set and let G be a group of permutations on that same set. Then G, a \ G is a subsemigroup, consisting of all non-invertible transformations, in the semigroup generated by G and a. Likewise, the conjugates a g = g −1 ag of a by elements g ∈ G generate a semigroup denoted a g | g ∈ G . We classify the finite permutation groups G on a finite set X such that the semigroups G, a , G, a \G, and a g | g ∈ G are regular for all transformations of X. We also classify the permutation groups G on a finite set X such that the semigroups G, a \ G and a g | g ∈ G are generated by their idempotents for all non-invertible transformations of X.