Abundant Rees matrix semigroups

Semigroup Forum, 2002

We give a necessary and sufficient condition for a locally inverse semigroup to be embeddable into a Rees matrix semigroup over a generalized inverse semigroup.

Portugaliae Mathematica, 2013

We introduce a class of finite semigroups obtained by considering Rees quotients of numerical semigroups. Several natural questions concerning this class, as well as particular subclasses obtained by considering some special ideals, are answered while others remain open. We exhibit nice presentations for these semigroups and prove that the Rees quotients by ideals of N, the positive integers under addition, constitute a set of generators for the pseudovariety of commutative and nilpotent semigroups.

Monatshefte f�r Mathematik, 2000

We formulate a general condition, called an enlargement, under which a semigroup T is covered by a Rees matrix semigroup over a subsemigroup.

Journal of Algebra, 2000

In "Semigroup Algebras," Okniński posed the following question: characterize semigroup algebras that are hereditary. In this paper we describe the (prime contracted) semigroup algebras K S that are hereditary and Noetherian when S is either a Malcev nilpotent monoid, a cancellative monoid or a monoid extension of a finite non-null Rees matrix semigroup. Furthermore, for the class of monoids which have an ideal series with factors that are non-null Rees matrix semigroups, we obtain an upper bound for the global dimension of its contracted semigroup algebra.

Journal of Combinatorial Theory, 1968

If M is a maximal (proper) subsemigroup of a finite semigroup S, then M contains all but one J-class J(M) of S. When J(M)is non-regular J(M)n M = ~ so M = S --J(M). When J(M) is regular either J(M) c3 M = ~ or M n J(M) has a natural form with respect to the Green-Rees coordinates in J(M). Specifically, there exist an isomorphism j : J(M) ~ ~ dl~ B, G, C) of J(M) ~ with a Rees regular matrix semigroup so that j(M c~ J(M)) = G" x A x B, where G' is a maximal subgroup of G or j(M n J(M)) is the complement of a "rectangle" of ,,~-classes of.Z/~ B, G, C). In the first case, (M c3 J(M)) o is a maximal subsemigroup of J(M) ~ In the second, (M n J(M)) o is maximal in J(M) ~ when j(M n J(M)) = r162176 B, G, C) --(G • A' x B ~) for proper subsets A' and B' of A and B, respectively, but need not be when j(Mn J(M)) = G X A X B' or j(M n J(M)) = G X A' X B. The notation of this paper, with slight variations, follows [1]. ~'~ B, G, C) denotes a Rees matrix semigroup with finite index sets A, B, finite group G and C : B • A ~ G o the structure matrix. If J is a J-class of a semigroup S, we denote by j0 the semigroup J w {0}, 0 r J, with multiplication tjlj~ ,

2017

Let S be a non trivial additive cancellation commutative semi-group having a zero 0 with the quotient group G = {s 1 − s 2 | s 1 , s 2 ∈ S}. A prime ideal P of S is called strongly prime if for a, b ∈ G, whenever a + b ∈ P , we get a ∈ P or b ∈ P. We show that a prime ideal of S is strongly prime ⇔ it is powerful. Finally, we prove that if O is an oversemigroup of S and S and O have the nonzero ideal I where I is powerful in O, then 3I is a powerful ideal of S.

Semigroup Forum, 1991

Acta Universitatis Apulensis, 2011

Let S = M o (G, P, I) be a Rees matrix semigroup with zero over a group G, we show that the approximate amenability of 1 (S) is equivalent to its amenability whenever the group G is amenable and the index set I is finite.

Computational and Mathematical Methods in Medicine

Semigroups are generalizations of groups and rings. In the semigroup theory, there are certain kinds of band decompositions which are useful in the study of the structure of semigroups. This research will open up new horizons in the field of mathematics by aiming to use semigroup ofh-bi-ideal of semiring with semilattice additive reduct. With the course of this research, it will prove that subsemigroup, the set of all righth-bi-ideals, and set of all lefth-bi-ideals are bands forh-regular semiring. Moreover, it will be demonstrated that if semigroup of allh-bi-idealsBH,∗is semilattice, thenHish-Clifford. This research will also explore the classification of minimalh-bi-ideal.

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...