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.

We establish a new sufficient condition under which a monoid is nonfinitely based and apply this condition to Lee monoids $L_{\ell }^{1}$, obtained by adjoining an identity element to the semigroup generated by two idempotents $a$ and $b$ with the relation $0=abab\cdots \,$ (length $\ell$). We show that every monoid $M$ which generates a variety containing $L_{5}^{1}$ and is contained in the variety generated by $L_{\ell }^{1}$ for some $\ell \geq 5$ is nonfinitely based. We establish this result by analysing $\unicode[STIX]{x1D70F}$-terms for $M$, where $\unicode[STIX]{x1D70F}$ is a certain nontrivial congruence on the free semigroup. We also show that if $\unicode[STIX]{x1D70F}$ is the trivial congruence on the free semigroup and $\ell \leq 5$, then the $\unicode[STIX]{x1D70F}$-terms (isoterms) for $L_{\ell }^{1}$ carry no information about the nonfinite basis property of $L_{\ell }^{1}$.

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.

The action of any group on itself by conjugation and the corresponding conjugacy relation play an important role in group theory. There have been many attempts to find notions of conjugacy in semigroups that would be useful in special classes of semigroups occurring in various areas of mathematics, such as semigroups of matrices, operator and topological semigroups, free semigroups, transition monoids for automata, semigroups given by presentations with prescribed properties, monoids of graph endomorphisms, etc. In this paper we study four notions of conjugacy for semigroups, their interconnections, similarities and dissimilarities. They appeared originally in various different settings (automata, representation theory, presentations, and transformation semigroups). Here we study them in full generality. The paper ends with a large list of open problems.

For each positive $n$, let $\mathbf{u}_{n}\approx \boldsymbol{v}_{n}$ denote the identity obtained from the Adjan identity $(xy)(yx)(xy)(xy)(yx)\approx (xy)(yx)(yx)(xy)(yx)$ by substituting $(xy)\rightarrow (x_{1}x_{2}\ldots x_{n})$ and $(yx)\rightarrow (x_{n}\ldots x_{2}x_{1})$. We show that every monoid which satisfies $\mathbf{u}_{n}\approx \boldsymbol{v}_{n}$ for each positive $n$ and generates a variety containing the bicyclic monoid is nonfinitely based. This implies that the monoid $U_{2}(\mathbb{T})$ (respectively, $U_{2}(\overline{\mathbb{Z}})$) of two-by-two upper triangular tropical matrices over the tropical semiring $\mathbb{T}=\mathbb{R}\cup \{-\infty \}$ (respectively, $\overline{\mathbb{Z}}=\mathbb{Z}\cup \{-\infty \}$) is nonfinitely based.

We give a sufficient condition under which a semigroup is nonfinitely based. As an application, we show that a certain variety is nonfinitely based, and we indicate the additional analysis (to be presented in a forthcoming paper), which shows that this example is a new limit variety of aperiodic monoids.

A paper of Almeida and Trotter [‘The pseudoidentity problem and reducibility for completely regular semigroups’, Bull. Aust. Math. Soc.63 (2001), 407–433] makes essential use of free profinite semigroupoids over profinite graphs with infinitely many vertices. It has since been shown that such structures must be handled with great care. In this note, it is verified that the required properties hold for the profinite graphs considered by Almeida and Trotter, thereby filling the gaps in the proof.

Let 𝒯n(F) denote the monoid of all upper triangular n×n matrices over a finite field F. It has been shown by Volkov and Goldberg that 𝒯n(F) is nonfinitely based if ∣F∣>2 and n≥4, but the cases when ∣F∣>2 and n=2,3 or when ∣F∣=2 have remained open. In this paper, it is shown that the monoid 𝒯2 (F) is finitely based when ∣F∣=2 , and a finite identity basis for it is given. Moreover, all maximal subvarieties of the variety generated by 𝒯2 (F) with ∣F∣=2 are determined.

Only three of the 15 973 distinct six-element semigroups are presently known to be nonfinitely based. This paper introduces a fourth example.

The operator that constructs the pseudovariety generated by the idempotent-generated semi-groups of a given pseudovariety is investigated. Several relevant examples of pseudovarieties generated by their idempotent-generated elements are given, as well as some properties of this operator. Particular attention is paid to the pseudovarieties in {J, R, L, DA} concerning this operator and their generator ranks and idempotent-generator ranks.

Completely regular semigroups CR are regarded here as algebras with multiplication and the unary operation of inversion. Their lattice of varieties is denoted by L(CR). Let B denote the variety of bands and L(B) the lattice of its subvarieties. The mapping V → V ∩ B is a complete homomorphism of L(CR) onto L(B). The congruence induced by it has classes that are intervals, say VB = [VB, VB] for V ∈ L(CR). Here VB = V ∩ B. We characterize VB in several ways, the principal one being an inductive way of constructing bases for v-irreducible band varieties. We term the latter canonical. We perform a similar analysis for the intersection of these varieties with the varieties BG, OBG and B.

The pseudovariety $\mathbf{DG}$ of all finite monoids all of whose regular $\mathcal{D}$-classes are subgroups is shown to be local, that is, it is verified that the pseudovariety $g\mathbf{DG}$ of finite categories generated by $\mathbf{DG}$ coincides with the pseudovariety $\ell\mathbf{DG}$ of all finite categories whose local monoids all belong to $\mathbf{DG}$. Yet more general statements of this kind are deduced, yielding results such as that, for every prime number $p$, the pseudovariety $\mathbf{DG}_p$ of all finite monoids all of whose regular $\mathcal{D}$-classes are $p$-groups is local, or that the pseudovarieties $\mathbf{DG}_{\mathrm{sol}}$ and $\mathbf{DG}_{\mathrm{nil}}$ of all finite monoids all of whose regular \mbox{$\mathcal{D}$-classes} are, respectively, solvable groups and nilpotent groups are local.

For a set $X$ and a variety $\mathcal{V}$ of bands, let $B_{\mathcal{V}}(X)$ be the relatively free band in $\mathcal{V}$ on $X$. For an arbitrary band variety $\mathcal{V}$ and an arbitrary set $X$, we determine the group of automorphisms of $\mathrm{End}(B_{\mathcal{V}}(X))$, the monoid of endomorphisms of $B_{\mathcal{V}}(X)$.

It has been proved by D. E. Cohen [1] that the lattice of all varieties of metabelian groups is countable. In this paper, we show that the lattice of all varieties of completely simple semigroups with metabelian subgroups has the cardinality of the continuum. M. Petrich and N. R. Reilly have introduced in [6] the notion of near varieties of idempotent generated completely simple semigroups. The mapping assigning to every variety $\mathcal{V}$ of completely simple semigroups the class of all idempotent generated members of $\mathcal{V}$ is a complete lattice homomorphism of the lattice of all varieties of completely simple semigroups onto the lattice of all near varieties of idempotent generated completely simple semigroups. In this paper we show that, in fact, the lattice of all near varieties of idempotent generated completely simple semigroups with metabelian subgroups has itself the cardinality of the continuum.