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.

IDENTITIES IN BRANDT SEMIGROUPS, REVISITED

We present a new proof for the main claim made in the author's paper "On the identity bases of Brandt semigroups" (Ural. Gos. Univ. Mat. Zap., 14, no.1 (1985), 38–42); this claim provides an identity basis for an arbitrary Brandt semigroup over a group of finite exponent. We also show how to fill a gap in the original proof of the claim in loc. cit.

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.

THE LATTICE OF VARIETIES OF STRICT LEFT RESTRICTION SEMIGROUPS

Left restriction semigroups are the unary semigroups that abstractly characterize semigroups of partial maps on a set, where the unary operation associates to a map the identity element on its domain. This paper is the sequel to two recent papers by the author, melding the results of the first, on membership in the variety $\mathbf{B}$ of left restriction semigroups generated by Brandt semigroups and monoids, with the connection established in the second between subvarieties of the variety $\mathbf{B}_{R}$ of two-sided restriction semigroups similarly generated and varieties of categories, in the sense of Tilson. We show that the respective lattices ${\mathcal{L}}(\mathbf{B})$ and ${\mathcal{L}}(\mathbf{B}_{R})$ of subvarieties are almost isomorphic, in a very specific sense. With the exception of the members of the interval $[\mathbf{D},\mathbf{D}\vee \mathbf{M}]$, every subvariety of $\mathbf{B}$ is induced from a member of $\mathbf{B}_{R}$ and vice versa. Here $\mathbf{D}$ is generated by the three-element left restriction semigroup $D$ and $\mathbf{M}$ is the variety of monoids. The analogues hold for pseudovarieties.

VARIETIES OF LEFT RESTRICTION SEMIGROUPS

Left restriction semigroups are the unary semigroups that abstractly characterize semigroups of partial maps on a set, where the unary operation associates to a map the identity element on its domain. They may be defined by a simple set of identities and the author initiated a study of the lattice of varieties of such semigroups, in parallel with the study of the lattice of varieties of two-sided restriction semigroups. In this work we study the subvariety $\mathbf{B}$ generated by Brandt semigroups and the subvarieties generated by the five-element Brandt inverse semigroup $B_{2}$, its four-element restriction subsemigroup $B_{0}$ and its three-element left restriction subsemigroup $D$. These have already been studied in the ‘plain’ semigroup context, in the inverse semigroup context (in the first two instances) and in the two-sided restriction semigroup context (in all but the last instance). The author has previously shown that in the last of these contexts, the behavior is pathological: ‘almost all’ finite restriction semigroups are inherently nonfinitely based. Here we show that this is not the case for left restriction semigroups, by exhibiting identities for the above varieties and for their joins with monoids (the analog of groups in this context). We do so by structural means involving subdirect decompositions into certain primitive semigroups. We also show that each identity has a simple structural interpretation.