Brazil et al. [‘Maximal subgroups of infinite symmetric groups’, Proc. Lond. Math. Soc. (3) 68(1) (1994), 77–111] provided a new family of maximal subgroups of the symmetric group $G(X)$ defined on an infinite set X. It is easy to see that, in this case, $G(X)$ contains subsemigroups that are not groups, but nothing is known about nongroup maximal subsemigroups of $G(X)$. We provide infinitely many examples of such semigroups.

The endomorphism monoid of a model-theoretic structure carries two interesting topologies: on the one hand, the topology of pointwise convergence induced externally by the action of the endomorphisms on the domain via evaluation; on the other hand, the Zariski topology induced within the monoid by (non-)solutions to equations. For all concrete endomorphism monoids of $\omega $-categorical structures on which the Zariski topology has been analysed thus far, the two topologies were shown to coincide, in turn yielding that the pointwise topology is the coarsest Hausdorff semigroup topology on those endomorphism monoids.

We establish two systematic reasons for the two topologies to agree, formulated in terms of the model-complete core of the structure. Further, we give an example of an $\omega $-categorical structure on whose endomorphism monoid the topology of pointwise convergence and the Zariski topology differ, answering a question of Elliott, Jonušas, Mitchell, Péresse, and Pinsker.

We investigate tameness of Toeplitz shifts. By introducing the notion of extended Bratteli–Vershik diagrams, we show that such shifts with finite Toeplitz rank are tame if and only if there are at most countably many orbits of singular fibres over the maximal equicontinuous factor. The ideas are illustrated using the class of substitution shifts. A body of elaborate examples shows that the assumptions of our results cannot be relaxed.

We show that, given a compact minimal system $(X,g)$ and an element h of the topological full group $\tau [g]$ of g, the infinite orbits of h admit a locally constant orientation with respect to the orbits of g. We use this to obtain a clopen partition of $(X,G)$ into minimal and periodic parts, where G is any virtually polycyclic subgroup of $\tau [g]$ . We also use the orientation of orbits to give a refinement of the index map and to describe the role in $\tau [g]$ of the submonoid generated by the induced transformations of g. Finally, we consider the problem, given a homeomorphism h of the Cantor space X, of determining whether or not there exists a minimal homeomorphism g of X such that $h \in \tau [g]$ .

Let V be an infinite-dimensional vector space over a field F and let $I(V)$ be the inverse semigroup of all injective partial linear transformations on V. Given $\alpha \in I(V)$ , we denote the domain and the range of $\alpha $ by ${\mathop {\textrm {dom}}}\,\alpha $ and ${\mathop {\textrm {im}}}\,\alpha $ , and we call the cardinals $g(\alpha )={\mathop {\textrm {codim}}}\,{\mathop {\textrm {dom}}}\,\alpha $ and $d(\alpha )={\mathop {\textrm {codim}}}\,{\mathop {\textrm {im}}}\,\alpha $ the ‘gap’ and the ‘defect’ of $\alpha $ . We study the semigroup $A(V)$ of all injective partial linear transformations with equal gap and defect and characterise Green’s relations and ideals in $A(V)$ . This is analogous to work by Sanwong and Sullivan [‘Injective transformations with equal gap and defect’, Bull. Aust. Math. Soc. 79 (2009), 327–336] on a similarly defined semigroup for the set case, but we show that these semigroups are never isomorphic.

In this article, we calculate the Ellis semigroup of a certain class of constant-length substitutions. This generalizes a result of Haddad and Johnson [IP cluster points, idempotents, and recurrent sequences. Topology Proc.22 (1997) 213–226] from the binary case to substitutions over arbitrarily large finite alphabets. Moreover, we provide a class of counterexamples to one of the propositions in their paper, which is central to the proof of their main theorem. We give an alternative approach to their result, which centers on the properties of the Ellis semigroup. To do this, we also show a new way to construct an almost automorphic–isometric tower to the maximal equicontinuous factor of these systems, which gives a more particular approach than the one given by Dekking [The spectrum of dynamical systems arising from substitutions of constant length. Z. Wahrscheinlichkeitstheor. Verw. Geb.41(3) (1977/78) 221–239].

Consider the action of $\operatorname{GL}(n,\mathbb{Q}_{p})$ on the $p$-adic unit sphere ${\mathcal{S}}_{n}$ arising from the linear action on $\mathbb{Q}_{p}^{n}\setminus \{0\}$. We show that for the action of a semigroup $\mathfrak{S}$ of $\operatorname{GL}(n,\mathbb{Q}_{p})$ on ${\mathcal{S}}_{n}$, the following are equivalent: (1) $\mathfrak{S}$ acts distally on ${\mathcal{S}}_{n}$; (2) the closure of the image of $\mathfrak{S}$ in $\operatorname{PGL}(n,\mathbb{Q}_{p})$ is a compact group. On ${\mathcal{S}}_{n}$, we consider the ‘affine’ maps $\overline{T}_{a}$ corresponding to $T$ in $\operatorname{GL}(n,\mathbb{Q}_{p})$ and a nonzero $a$ in $\mathbb{Q}_{p}^{n}$ satisfying $\Vert T^{-1}(a)\Vert _{p}<1$. We show that there exists a compact open subgroup $V$, which depends on $T$, such that $\overline{T}_{a}$ is distal for every nonzero $a\in V$ if and only if $T$ acts distally on ${\mathcal{S}}_{n}$. The dynamics of ‘affine’ maps on $p$-adic unit spheres is quite different from that on the real unit spheres.

The set of row reduced matrices (and of echelon form matrices) is closed under multiplication. We show that any system of representatives for the $\text{Gl}_{n}(\mathbb{K})$ action on the $n\times n$ matrices, which is closed under multiplication, is necessarily conjugate to one that is in simultaneous echelon form. We call such closed representative systems Grassmannian semigroups. We study internal properties of such Grassmannian semigroups and show that they are algebraic semigroups and admit gradings by the finite semigroup of partial order preserving permutations, with components that are naturally in one–one correspondence with the Schubert cells of the total Grassmannian. We show that there are infinitely many isomorphism types of such semigroups in general, and two such semigroups are isomorphic exactly when they are semiconjugate in $M_{n}(\mathbb{K})$. We also investigate their representation theory over an arbitrary field, and other connections with multiplicative structures on Grassmannians and Young diagrams.

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.

We characterise the elements of the (maximum) idempotent-generated subsemigroup of the Kauffman monoid in terms of combinatorial data associated with certain normal forms. We also calculate the smallest size of a generating set and idempotent generating set.

We augment the body of existing results on embedding finite semigroups of a certain type into 2-generator finite semigroups of the same type. The approach adopted applies to finite semigroups the idempotents of which form a band and also to finite orthodox semigroups.

Non-degenerate monoids of skew type are considered. This is a class of monoids S defined by n generators and quadratic relations of certain type, which includes the class of monoids yielding set-theoretic solutions of the quantum Yang–Baxter equation, also called binomial monoids (or monoids of I-type with square-free defining relations). It is shown that under any degree-lexicographic order on the associated free monoid FMn. of rank n the set of normal forms of elements of S is a regular language in FMn. As one of the key ingredients of the proof, it is shown that an identity of the form xN yN = yN xN holds in S. The latter is derived via an investigation of the structure of S viewed as a semigroup of matrices over a field. It also follows that the semigroup algebra K[S] is a finite module over a finitely generated commutative subalgebra of the form K[A] for a submonoid A of S.

Let 𝒯X be the full transformation semigroup on a set X and E be a nontrivial equivalence on X. Write then TE(X) is a subsemigroup of 𝒯X. In this paper, we endow TE(X) with the so-called natural order and determine when two elements of TE(X) are related under this order, then find out elements of TE(X) which are compatible with ≤ on TE(X). Also, the maximal and minimal elements and the covering elements are described.

In this paper we introduce the notion of normally ordered block-group as a natural extension of the notion of normally ordered inverse semigroup considered previously by the author. We prove that the class NOS of all normally ordered block-groups forms a pseudovariety of semigroups and, by using the Munn representation of a block-group, we deduce the decompositions in Mal'cev products NOS = EIPOI and NOS ∩ A = NPOI, where A, EI and N denote the pseudovarieties of all aperiodic semigroups, all semigroups with just one idempotent and all nilpotent semigroups, respectively, and POI denotes the pseudovariety of semigroups generated by all semigroups of injective order-preserving partial transformations on a finite chain. These relations are obtained after showing the equalities BG = EIEcom = NEcom, where BG and Ecom denote the pseudovarieties of all block-groups and all semigroups with commuting idempotents, respectively.

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 $B_{\mathcal{V}}(X)$.

The partial automorphism monoid of an inverse semigroup is an inverse monoid consisting of all isomorphisms between its inverse subsemigroups. We prove that a tightly connected fundamental inverse semigroup S with no isolated nontrivial subgroups is lattice determined ‘modulo semilattices’ and if T is an inverse semigroup whose partial automorphism monoid is isomorphic to that of S, then either S and T are isomorphic or they are dually isomorphic chains relative to the natural partial order; a similar result holds if T is any semigroup and the inverse monoids consisting of all isomorphisms between subsemigroups of S and T, respectively, are isomorphic. Moreover, for these results to hold, the conditions that S be tightly connected and have no isolated nontrivial subgroups are essential.

Let $n$ and $r$ be positive integers with $1\,{<}\,r\,{<}\,n$, and let $X_n\,{=}\break\{1,2,\ldots,n\}$. An $r$-set $A$ and a partition $\pi$ of $X_n$ are said to be orthogonal if every class of $\pi$ meets $A$ in exactly one element. We prove that if $A_{1},A_{2},\ldots, A_{\binom n r}$ is a list of the distinct $r$-sets of $X_ n$ with $|A_{i}\cap A_{i+1}|\,{=}\,r-1$ for $i=1,2,\ldots, \binom n r$ taken modulo $\binom n r$, then there exists a list of distinct partitions $\pi_{1},\pi_{2},\ldots, \pi_{\binom n r}$ such that $\pi_{i}$ is orthogonal to both $A_{i}$ and $A_{i+1}$. This result states that any constant weight Gray code admits a labeling by distinct orthogonal partitions. Using an algorithm from the literature on Gray codes, we provide a surprisingly efficient algorithm that on input $(n,r)$ outputs an orthogonally labeled constant weight Gray code. We also prove a two-fold Gray enumeration result, presenting an orthogonally labeled constant weight Gray code in which the partition labels form a cycle in the covering graph of the lattice of all partitions of $X_n$. This leads to a conjecture related to the Middle Levels Conjecture. Finally, we provide an application of our results to calculating minimal generating sets of idempotents for finite semigroups.

This paper is mainly dedicated to describing the congruences on certain monoids of transformations on a finite chain $X_n$ with $n$ elements. Namely, we consider the monoids $\od_n$ and $\mpod_n$ of all full, respectively partial, transformations on $X_n$ that preserve or reverse the order, as well as the submonoid $\po_n$ of $\mpod_n$ of all its order-preserving elements. The inverse monoid $\podi_n$ of all injective elements of $\mpod_n$ is also considered.

We show that in $\po_n$ any congruence is a Rees congruence, but this may not happen in the monoids $\od_n$, $\podi_n$ and $\mpod_n$. However in all these cases the congruences form a chain.

In this paper we consider collections of compact operators on a real or complex Banach space including linear operators on finite-dimensional vector spaces. We show that such a collection is simultaneously triangularizable if and only if it is arbitrarily close to a simultaneously triangularizable collection of compact operators. As an application of these results we obtain an invariant subspace theorem for certain bounded operators. We further prove that in finite dimensions near reducibility implies reducibility whenever the ground field is $\mathbb{R}$ or $\mathbb{C}$.