Coherency and Constructions for Monoids

A 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.

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.

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 $n\geq 2$.

HOMOMORPHISMS OF -MUNN ALGEBRAS AND APPLICATIONS TO SEMIGROUP ALGEBRAS

In this paper, for an arbitrary$\ell ^{1}$-Munn algebra$\mathfrak{A}$over a Banach algebra$A$with a sandwich matrix$P$, we characterise all homomorphisms from$\mathfrak{A}$to a commutative Banach algebra$B$. Especially, we study the character space of this algebra. Then, as an application, its character amenability is investigated. Finally, we apply these results to certain semigroups, which are called Rees matrix semigroups.

OPTIMAL REES MATRIX CONSTRUCTIONS FOR ANALYSIS OF DATA

We introduce a new construction involving Rees matrix semigroups and max-plus algebras that is very convenient for generating sets of centroids. We describe completely all optimal sets of centroids for all Rees matrix semigroups without any restrictions on the sandwich matrices.

COMPLETELY REGULAR MONOIDS WITH TWO GENERATORS

We classify semigroups in the title according to whether they have a finite or an infinite number ofℒ-classes or ℛ-classes. For each case, we provide a concrete construction using Rees matrix semigroups and their translational hulls. An appropriate relatively free semigroup is used to complete the classification. All this is achieved by first treating the special case in which one of the generators is idempotent. We conclude by a discussion of a possible classification of 2-generator completely regular semigroups.

REES MATRIX CONSTRUCTIONS FOR CLUSTERING OF DATA

This paper continues the investigation of semigroup constructions motivated by applications in data mining. We give a complete description of the error-correcting capabilities of a large family of clusterers based on Rees matrix semigroups well known in semigroup theory. This result strengthens and complements previous formulas recently obtained in the literature. Examples show that our theorems do not generalize to other classes of semigroups.