Adequate Semigroups (original) (raw)
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Proper left type-A monoids revisited
Glasgow Mathematical Journal, 1993
The relation ℛ* is defined on a semigroup S by the rule that ℛ*b if and only if the elements a, b of S are related by the Green's relation ℛ in some oversemigroup of S. A semigroup S is an E-semigroup if its set E(S)of idempotents is a subsemilattice of S. A left adequate semigroup is an E-semigroup in which every ℛ*-class contains an idempotent. It is easy to see that, in fact, each ℛ*-class of a left adequate semigroup contains a unique idempotent [2]. We denote the idempotent in the ℛ*-class of a by a+.
Left Adequate and Left Ehresmann Monoids
International Journal of Algebra and Computation, 2011
This is the first of two articles studying the structure of left adequate and, more generally, of left Ehresmann monoids. Motivated by a careful analysis of normal forms, we introduce here a concept of proper for a left adequate monoid M . In fact, our notion is that of T -proper, where T is a submonoid of M . We show that any left adequate monoid M has an X *proper cover for some set X, that is, there is a left adequate monoid M that is X * -proper, and an idempotent separating epimorphism θ : M → M of the appropriate type. Given this result, we may deduce that the free left adequate monoid on any set X is X * -proper.
In this paper the terms; a completely prime po-Γ-ideal, c-system,a prime po-Γ-ideal, m-system of a po--semigroup are introduced. It is proved that every po--subsemigroup of a po--semigroup is a c-system. It is also proved that a po--ideal P of a po--semigroup S is completely prime if and only if S\P is either a c-system or empty. It is proved that if P is a po-Γ-ideal of a po-Γ-semigroup S, then the conditions if A, B are po-Γ-ideals of S and AΓB⊆P then either A⊆P or B⊆P, (2) if a, b ∈ S such that aΓS 1 Γb ⊆ P, then either a ∈ P or b ∈ P, are equivalent.
Studying Semigroups Using the Properties of Their Prime m-Ideals
The Bulletin of Irkutsk State University, Series, Mathematics, 2020
In this article, we present the idea of m-ideals, prime m-ideals and their associated types for a positive integer m in a semigroup. We present different chrarc-terizations of semigroups through m-ideals. We demonstrate that the ordinary ideals, and their relevent types differ from the m-ideals and their assocated types by presenting concrete examples on the maximal, irreducible and strongly irreducible m-ideals. We conclude from the study that the introduction of the m-ideal will explore new fields of studies in semigroups and their applications.
IDEALS IN LEFT ALMOST SEMIGROUPS
A left almost semigroup (LA-semigroup) or an Abel-Grassmann's groupoid (AG-groupoid) is investigated in several papers. In this paper we have discussed ideals in LA-semigroups. Specifically, we have shown that every ideal in an LA-semigroup S with left identity e is prime if and only if it is idempotent and the set of ideals of S is totally ordered under inclusion. We have shown that an ideal of S is prime if and only if it is semiprime and strongly irreducible. We have proved also that every ideal in a regular LA-semigroup S is prime if and only if the set of ideals of S is totally ordered under inclusion. We have proved in the end that every ideal in S is prime if and only if it is strongly irreducible and the set of ideals of S form a semilattice. 2000 Mathematics Subject Classification. 20M10 and 20N99.
Journal of Algebra, 2005
Right chain semigroups are semigroups in which right ideals are linearly ordered by inclusion. Multiplicative semigroups of right chain rings, right cones, right invariant right holoids and right valuation semigroups are examples. The ideal theory of right chain semigroups is described in terms of prime and completely prime ideals, and a classification of prime segments is given, extending to these semigroups results on right cones proved by Brungs and Törner [H.H. Brungs, G. Törner, Ideal theory of right cones and associated rings, J. Algebra 210 (1998) 145-164].