Rees Matrix Semigroups Research Articles

On Rees matrix semigroups M(S;I,Λ;P) over semigroups S in which I is a singleton

Let M ( S ; Λ ; P ) denote a Rees I × Λ matrix semigroup over a semigroup S in which I is a singleton. In this paper we describe the right regular representation of M ( S ; Λ ; P ) , and find connection between properties of S and M ( S ; Λ ; P ) . We also prove embedding theorems on Rees matrix semigroups M ( S ; Λ ; P ) .

On special Rees matrix semigroups over semigroups

On special Rees matrix semigroups over semigroups

Representations of Munn algebras and related semigroups

We establish representation types (finite, tame or wild) of finite dimensional Munn algebras with semisimple bases. As an application, we establish representation types of finite Rees matrix semigroups, in particular, 0-simple semigroups, and their mutually annihilating unions.

Rees matrix seminearring

As a continuation of the work done in (R. Mukherjee (Pal), P. Pal and S. K. Sardar, On additively completely regular seminearrings, Commun. Algebra 45 (12) (2017) 5111–5122), in this paper, our objective is to characterize left (right) completely simple seminearrings in terms of Rees Construction by generalizing the concept of Rees matrix semigroup (J. M. Howie, Fundamentals of Semigroup Theory (Clarendon Press, Oxford, 1995); M. Petrich and N. R. Reilly, Completely Regular Semigroups (Wiley, New York, 1999)) and that of Rees matrix semiring (M. K. Sen, S. K. Maity and H. J. Weinert, Completely simple semirings, Bull. Calcutta Math. Soc. 97 (2005) 163–172). In Rees theorem, a completely simple semigroup is coordinatized in such a way that each element can be seen to be a triplet which gives this abstract structure a much more simpler look. In this paper, we have been able to construct a similar kind of coordinate structure of a restricted class of left (right) completely simple seminearrings taking impetus from (M. P. Grillet, Semirings with a completely simple additive semigroup, J. Austral. Math. Soc. 20 (Ser. A) (1975) 257–267, Theorem [Formula: see text] and (M. K. Sen, S. K. Maity and H. J. Weinert, Completely simple semirings, Bull. Calcutta Math. Soc. 97 (2005) 163–172, Theorem 3.1).

Козображення матричних алгебр Манна

Let A be an algebra over a field K, m and n natural numbers and P = (pji) a fixed n x m matrix over A. The K-vector space of all m x n matrices over the algebra A can be made into an algebra with respect to the following operation (o): B o C = BPC. This algebra is called the Munn matrix algebra over A with sandwich matrix P. The algebras of such type arose as generalizations of semigroup algebras of Rees matrix semigroups which in turn are closely related to simple semigroups. This article describes the generators and defining relations of Mann matrix algebras with a regular sandwich matrix.

Projection-primitive $ P $-Ehresmann semigroups

$ P $-Ehresmann semigroups introduced by Jones are natural generalizations of Ehresmann semigroups. The aim of this paper is to introduce and describe projection-primitive $ P $-Ehresmann semigroups. More specifically, it is proved that a projection-primitive $ P $-Ehresmann semigroup is either isomorphic to a Rees matrix semigroup over a monoid with some natural conditions or a generalized category with zero adjoined.

Coherency and Constructions for Monoids

AbstractA monoid S is right coherent if every finitely generated subact of every finitely presented right S-act is finitely presented. This is a finiteness condition, and we investigate whether or not it is preserved under some standard algebraic and semigroup theoretic constructions: subsemigroups, homomorphic images, direct products, Rees matrix semigroups, including Brandt semigroups, and Bruck–Reilly extensions. We also investigate the relationship with the property of being weakly right noetherian, which requires all right ideals of S to be finitely generated.

Homological dimension of Rees matrix semigroups in intelligent fuzzy systems

Homological dimension of Rees matrix semigroups in intelligent fuzzy systems

Flows on Classes of Regular Semigroups and Cauchy Categories

We consider the structure of the flow monoid for some classes of regular semigroups (which are special case of flows on categories) and for Cauchy categories. In detail, we characterize flows for Rees matrix semigroups, rectangular bands, and full transformation semigroups and also describe the Cauchy categories for some classes of regular semigroups such as completely simple semigroups, Brandt semigroups, and rectangular bands. In fact, we obtain a general structure for the flow monoids on Cauchy categories.

Semigroups with finitely generated universal left congruence

We consider semigroups such that the universal left congruence omega ^{ell } is finitely generated. Certainly a left noetherian semigroup, that is, one in which all left congruences are finitely generated, satisfies our condition. In the case of a monoid the condition that omega ^{ell } is finitely generated is equivalent to a number of pre-existing notions. In particular, a monoid satisfies the homological finiteness condition of being of type left-{text {FP}}_1 exactly when omega ^{ell } is finitely generated.Our investigations enable us to classify those semigroups such that omega ^{ell } is finitely generated that lie in certain important classes, such as strong semilattices of semigroups, inverse semigroups, Rees matrix semigroups (over semigroups) and completely regular semigroups. We consider closure properties for the class of semigroups such that omega ^{ell } is finitely generated, including under morphic image, direct product, semi-direct product, free product and 0-direct union. Our work was inspired by the stronger condition, stated for monoids in the work of White, of being pseudo-finite. Where appropriate, we specialise our investigations to pseudo-finite semigroups and monoids. In particular, we answer a question of Dales and White concerning the nature of pseudo-finite monoids.

ON SEMIGROUPS WITH PSPACE-COMPLETE SUBPOWER MEMBERSHIP PROBLEM

Fix a finite semigroup $S$ and let $a_{1},\ldots ,a_{k},b$ be tuples in a direct power $S^{n}$. The subpower membership problem (SMP) for $S$ asks whether $b$ can be generated by $a_{1},\ldots ,a_{k}$. For combinatorial Rees matrix semigroups we establish a dichotomy result: if the corresponding matrix is of a certain form, then the SMP is in P; otherwise it is NP-complete. For combinatorial Rees matrix semigroups with adjoined identity, we obtain a trichotomy: the SMP is either in P, NP-complete, or PSPACE-complete. This result yields various semigroups with PSPACE-complete SMP including the six-element Brandt monoid, the full transformation semigroup on three or more letters, and semigroups of all $n$ by $n$ matrices over a field for $n\geq 2$.

On the Construction of Molaei’s Generalized Hypergroups

The purpose of this paper is making a construction and generalization of Molaei’s generalized groups by using construction of the Rees matrix semigroup over a polygroup H and a matrix with entries in H. We call it “Molaei’s generalized hypergroups” and we give some examples.

On the star-height of subword counting languages and their relationship to Rees zero-matrix semigroups

Given a word w over a finite alphabet, we consider, in three special cases, the generalised star-height of the languages in which w occurs as a contiguous subword (factor) an exact number of times and of the languages in which w occurs as a contiguous subword modulo a fixed number, and prove that in each case it is at most one. We use these combinatorial results to show that any language recognised by a Rees (zero-)matrix semigroup over an Abelian group is of generalised star-height at most one.

Cross-connections of completely simple semigroups

A completely simple semigroup [Formula: see text] is a semigroup without zero which has no proper ideals and contains a primitive idempotent. It is known that [Formula: see text] is a regular semigroup and any completely simple semigroup is isomorphic to the Rees matrix semigroup [Formula: see text] (cf. D. Rees, On semigroups, Proc. Cambridge Philos. Soc. 36 (1940) 387–400). In the study of structure theory of regular semigroups, Nambooripad introduced the concept of normal categories to construct the semigroup from its principal left (right) ideals using cross-connections. A normal category [Formula: see text] is a small category with subobjects wherein each object of the category has an associated idempotent normal cone and each morphism admits a normal factorization. A cross-connection between two normal categories [Formula: see text] and [Formula: see text] is a local isomorphism [Formula: see text] where [Formula: see text] is the normal dual of the category [Formula: see text]. In this paper, we identify the normal categories associated with a completely simple semigroup [Formula: see text] and show that the semigroup of normal cones [Formula: see text] is isomorphic to a semi-direct product [Formula: see text]. We characterize the cross-connections in this case and show that each sandwich matrix [Formula: see text] correspond to a cross-connection. Further we use each of these cross-connections to give a representation of the completely simple semigroup as a cross-connection semigroup.

THE CONGRUENCE η* ON SEMIGROUPS

In this paper we define a congruence $\eta^{\ast}$ on semigroups. For the finite semigroups $S$, $\eta^{\ast}$ is the smallest congruence relation such that $S/\eta^{\ast}$ is a nilpotent semigroup (in the sense of Malcev). In order to study the congruence relation $\eta^{\ast}$ on finite semigroups, we define a $\textbf{CS}$-diagonal finite regular Rees matrix semigroup. We prove that, if $S$ is a $\textbf{CS}$-diagonal finite regular Rees matrix semigroup then $S/\eta^{\ast}$ is inverse. Also, if $S$ is a completely regular finite semigroup, then $S/\eta^{\ast}$ is a Clifford semigroup. We show that, for every non-null principal factor $A/B$ of $S$, there is a special principal factor $C/D$ such that every element of $A\setminus B$ is $\eta^{\ast}$-equivalent with some element of $C\setminus D$. We call the principal factor $C/D$, the $\eta^{\ast}$-root of $A/B$. All $\eta^{\ast}$-roots are $\textbf{CS}$-diagonal. If certain elements of $S$ act in the special way on the $\textbf{R}$-classes of a $\textbf{CS}$-diagonal principal factor then it is not an $\eta^{\ast}$-root. Some of these results are also expressed in terms of pseudovarieties of semigroups.

On the Cayley Graphs of Completely Simple Semigroups

In this paper first we study and characterize the Cayley graph of Rees matrix semigroups (completely simple semigroups). Then we investigate Cayley D-saturated property of these graphs and we study the Cayley D-saturated property of Rees matrix semigroups.

David Rees, FRS 1918–2013

David Rees, FRS 1918–2013

Homomorphisms of $$\ell ^1$$ ℓ 1 -Munn algebras and Rees matrix semigroup algebras

Let \({\mathcal {LM}}\left( {\mathcal {A}}, P\right) \) be an \(\ell ^1\)-Munn algebra over an arbitrary unital Banach algebra \({\mathcal {A}}\). We characterize homomorphisms from \({\mathcal {LM}}\left( {\mathcal {A}}, P\right) \) into an arbitrary Banach algebra \({\mathcal {B}}\) in terms of homomorphisms from \({\mathcal {A}}\) into \({\mathcal {B}}\). Then we discuss homomorphisms from arbitrary Banach algebras into \({\mathcal {LM}}\left( {\mathcal {A}}, P\right) \). Existence and uniqueness of homomorphisms under certain conditions are also discussed. We apply these results to the concrete case of \(\ell ^1(S)\) where S is a Rees matrix semigroup, to identify characters of \(\ell ^1(S)\) in both cases where S is with or without zero. As a consequence if the sandwich matrix of S has a zero entry, then \(\ell ^1(S)\) is character amenable.

Cellularity of Some Semigroup Algebras

In this paper, we study the cellularity of some semigroup algebras. We show that the semigroup algebra of a U-semiabundant semigroup with Rees matrix semigroups over monoids as its principal \(\sim _U\)-factors is a cellular algebra if and only if all of the monoid algebras are cellular. We also study the cellularity of the semigroup algebra of a semilattice of Rees matrix semigroups. As consequences, we get the cellularity of super abundant semigroup algebras and complete regular semigroup algebras.

Free completely J (ℓ)-simple Semigroups

A semigroup is called completely J(l)-simple if it is isomorphic to some Rees matrix semigroup over a left cancellative monoid and each entry of whose sandwich matrix is in the group of units of the left cancellative monoid. It is proved that completely J(l)-simple semigroups form a quasivarity. Moreover, the construction of free completely J(l)-simple semigroups is given. It is found that a free completely J(l)-simple semigroup is just a free completely J*-simple semigroup and also a full subsemigroup of some completely simple semigroups.