# Regular semigroups

About: This article is published in Semigroup Forum.The article was published on 1989-12-01. It has received 9 citations till now.

##### Citations

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TL;DR: In this article, the authors studied the structure of the partial Brauer monoid and its planar sub-monoid, the Motzkin monoid, and obtained necessary and sufficient conditions under which the ideals of these monoids are idempotent-generated.

43 citations

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TL;DR: In this article, the Munn semigroup of a semilattice E(P) is considered and the set of all C-isomorphisms from ⟨q⟩ onto ⟩ p is denoted by C q,p.

Abstract: Let E(P) be a 𝒫-regular partial band and 𝒰 = {(q, p) ∈ P × P ∣ ⟨q⟩ ≃ C⟨p⟩}. For any (q, p) ∈ 𝒰, the set of all C-isomorphisms from ⟨q⟩ onto ⟨p⟩ is denoted by C q,p . The set C = ∪ {C q,p ∣ (q, p) ∈ 𝒰} forms a 𝒫-regular semigroup with P + = {θ p ∣ ⟨q⟩ ∣ p ∈ V P (q)} as its C-set. Because it is a generalization of the Munn semigroup of a semilattice, C is referred to as the Munn semigroup of E(P). The properties of C(P +) and the connection between C(P +) and the Hall semigroup A(P*) of E(P) are discussed. Each 𝒫-regular semigroup is reconstructed with its greatest regular *-semigroup homomorphic image and the Munn semigroup of its 𝒫-regular partial band. This result generalizes results on orthodox semigroups. Noting also that C is a regular *-semigroup with P # = {I ⟨p⟩ ∣ p ∈ P} as the set of projections, a new structural description for strongly 𝒫-regular semigroups is obtained. This description only involves regular *-semigroups and avoids 𝒫-regular semigroups.

15 citations

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TL;DR: In this article, the authors give a characterisation of the P-kernel of a P-congruence and show that the class of regular *-semigroups are within the same class of P-regular semigroups.

Abstract: A pair (S, P) of a regular semigroupsS and a subsetP ofE
s
whereE
s
is the set of all idempotent elements ofS is called aP-regular semigroupS(P) if it satisfies the following:
The class of orthodox semigroups and the class of regular *-semigroups are within the class ofP-regular semigroups. This paper gives a characterisation of theP-kernel of aP-congruence.

7 citations

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TL;DR: In this article, the equivalence of a semigroup S in terms of a set U of idempotents in S is defined, and a class of U-liberal semigroups is characterized and some special cases are considered.

Abstract: In this paper the equivalence
$$\tilde {\cal Q}^U $$
on a semigroup S in terms of a set U of idempotents in S is defined. A semigroup S is called a U-liberal semigroup with U as the set of projections and denoted by S(U) if every
$$\tilde {\cal Q}^U $$
-class in it contains an element in U. A class of U-liberal semigroups is characterized and some special cases are considered.

5 citations

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TL;DR: In this paper, the authors investigated the properties and structure of regular *-transversals in regular semigroups and gave two kinds of definitions for regular * -transversal and proved that they are equivalent.

Abstract: Our aim is to investigate the properties and structure of the ℘-regular semigroups having regular *-transversals. We first give two kinds of definitions for regular *-transversals and prove that they are equivalent. A condition for two regular *-transversals of a ℘-regular semigroup to be isomorphic is given. The *-congruences on the ℘-regular semigroups having regular *-transversals are characterized by using the *-congruence triples. In particular, we describe the idempotent separating congruences and the group congruences on such ℘-regular semigroups.

4 citations

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TL;DR: In this article, it was shown that a regular semigroup S becomes a regular *-semigroup (in the sense of [1]) if and only if S has a certain subset called a p-system.

Abstract: In this paper, firstly it is shown that a regular semigroup S becomes a regular *-semigroup (in the sense of [1]) if and only if S has a certain subset called a p-system. Secondly, all the normal *-bands are completely described in terms of rectangular *-bands (square bands) and transitive systems of homomorphisms of rectangular *-bands. Further, it is shown that an orthodox semigroup S becomes a regular *-semigroup if there is a p-system F of the band ES of idempotents of S such that F∋e, ES∋t, e≥t imply t∈F. By using this result, it is also shown that F is a p-system of a generalized inverse semigroup S if and only if F is a p-system of FS.

16 citations