Motivated by approaches to the word problem for one-relation monoids arising from work of Adian and Oganesian (1987), Guba (1997), and Ivanov, Margolis, and Meakin (2001), we study the submonoid and rational subset membership problems in one-relation monoids and in positive one-relator groups. We give the first known examples of positive one-relator groups with undecidable submonoid membership problem, and we apply this to give the first known examples of one-relation monoids with undecidable submonoid membership problem. We construct several infinite families of one-relation monoids with undecidable submonoid membership problem, including examples that are defined by relations of the form $w=1$ but which are not groups, and examples defined by relations of the form $u=v$ where both of u and v are nonempty. As a consequence, we obtain a classification of the right-angled Artin groups that can arise as subgroups of one-relation monoids. We also give examples of monoids with a single defining relation of the form $aUb = a$ and examples of the form $aUb=aVa$, with undecidable rational subset membership problem. We give a one-relator group defined by a freely reduced word of the form $uv^{-1}$ with $u, v$ positive words, in which the prefix membership problem is undecidable. Finally, we prove the existence of a special two-relator inverse monoid with undecidable word problem, and in which both the relators are positive words. As a corollary, we also find a positive two-relator group with undecidable prefix membership problem. In proving these results, we introduce new methods for proving undecidability of the rational subset membership problem in monoids and groups, including by finding suitable embeddings of certain trace monoids.

The monogenic free inverse semigroup $FI_1$ is not finitely presented as a semigroup due to the classic result by Schein (1975). We extend this result and prove that a finitely generated subsemigroup of $FI_1$ is finitely presented if and only if it contains only finitely many idempotents. As a consequence, we derive that an inverse subsemigroup of $FI_1$ is finitely presented as a semigroup if and only if it is a finite semilattice.

Let $\mathbb {Z}$ be the additive (semi)group of integers. We prove that for a finite semigroup S the direct product $\mathbb {Z}\times S$ contains only countably many subdirect products (up to isomorphism) if and only if S is regular. As a corollary we show that $\mathbb {Z}\times S$ has only countably many subsemigroups (up to isomorphism) if and only if S is completely regular.

A set of complex numbers $S$ is called invariant if it is closed under addition and multiplication, namely, for any $x, y \in S$ we have $x+y \in S$ and $xy \in S$. For each $s \in {\mathbb {C}}$ the smallest invariant set ${\mathbb {N}}[s]$ containing $s$ consists of all possible sums $\sum _{i \in I} a_i s^i$, where $I$ runs over all finite nonempty subsets of the set of positive integers ${\mathbb {N}}$ and $a_i \in {\mathbb {N}}$ for each $i \in I$. In this paper, we prove that for $s \in {\mathbb {C}}$ the set ${\mathbb {N}}[s]$ is everywhere dense in ${\mathbb {C}}$ if and only if $s \notin {\mathbb {R}}$ and $s$ is not a quadratic algebraic integer. More precisely, we show that if $s \in {\mathbb {C}} \setminus {\mathbb {R}}$ is a transcendental number, then there is a positive integer $n$ such that the sumset ${\mathbb {N}} t^n+{\mathbb {N}} t^{2n} +{\mathbb {N}} t^{3n}$ is everywhere dense in ${\mathbb {C}}$ for either $t=s$ or $t=s+s^2$. Similarly, if $s \in {\mathbb {C}} \setminus {\mathbb {R}}$ is an algebraic number of degree $d \ne 2, 4$, then there are positive integers $n, m$ such that the sumset ${\mathbb {N}} t^n+{\mathbb {N}} t^{2n} +{\mathbb {N}} t^{3n}$ is everywhere dense in ${\mathbb {C}}$ for $t=ms+s^2$. For quadratic and some special quartic algebraic numbers $s$ it is shown that a similar sumset of three sets cannot be dense. In each of these two cases the density of ${\mathbb {N}}[s]$ in ${\mathbb {C}}$ is established by a different method: for those special quartic numbers, it is possible to take a sumset of four sets.

The complex algebra of an inverse semigroup with finitely many idempotents in each $\mathcal D$-class is stably finite by a result of Munn. This can be proved fairly easily using $C^{*}$-algebras for inverse semigroups satisfying this condition that have a Hausdorff universal groupoid, or more generally for direct limits of inverse semigroups satisfying this condition and having Hausdorff universal groupoids. It is not difficult to see that a finitely presented inverse semigroup with a non-Hausdorff universal groupoid cannot be a direct limit of inverse semigroups with Hausdorff universal groupoids. We construct here countably many nonisomorphic finitely presented inverse semigroups with finitely many idempotents in each $\mathcal D$-class and non-Hausdorff universal groupoids. At this time, there is not a clear $C^{*}$-algebraic technique to prove these inverse semigroups have stably finite complex algebras.

We investigate the groups of units of one-relator and special inverse monoids. These are inverse monoids which are defined by presentations, where all the defining relations are of the form $r=1$. We develop new approaches for finding presentations for the group of units of a special inverse monoid, and apply these methods to give conditions under which the group admits a presentation with the same number of defining relations as the monoid. In particular, our results give sufficient conditions for the group of units of a one-relator inverse monoid to be a one-relator group. When these conditions are satisfied, these results give inverse semigroup theoretic analogues of classical results of Adjan for one-relator monoids, and Makanin for special monoids. In contrast, we show that in general these classical results do not hold for one-relator and special inverse monoids. In particular, we show that there exists a one-relator special inverse monoid whose group of units is not a one-relator group (with respect to any generating set), and we show that there exists a finitely presented special inverse monoid whose group of units is not finitely presented.

A prefix monoid is a finitely generated submonoid of a finitely presented group generated by the prefixes of its defining relators. Important results of Guba (1997), and of Ivanov, Margolis and Meakin (2001), show how the word problem for certain one-relator monoids, and inverse monoids, can be reduced to solving the membership problem in prefix monoids of certain one-relator groups. Motivated by this, in this paper, we study the class of prefix monoids of finitely presented groups. We obtain a complete description of this class of monoids. All monoids in this family are finitely generated, recursively presented and group-embeddable. Our results show that not every finitely generated recursively presented group-embeddable monoid is a prefix monoid, but for every such monoid, if we take a free product with a suitably chosen free monoid of finite rank, then we do obtain a prefix monoid. Conversely, we prove that every prefix monoid arises in this way. Also, we show that the groups that arise as groups of units of prefix monoids are precisely the finitely generated recursively presented groups, whereas the groups that arise as Schützenberger groups of prefix monoids are exactly the recursively enumerable subgroups of finitely presented groups. We obtain an analogous result classifying the Schützenberger groups of monoids of right units of special inverse monoids. We also give some examples of right cancellative monoids arising as monoids of right units of finitely presented special inverse monoids, and we show that not all right cancellative recursively presented monoids belong to this class.

An example of a nonfinitely based involution monoid of order five has recently been discovered. We confirm that this example is, up to isomorphism, the unique smallest among all involution monoids.

This article studies the properties of word-hyperbolic semigroups and monoids, that is, those having context-free multiplication tables with respect to a regular combing, as defined by Duncan and Gilman [‘Word hyperbolic semigroups’, Math. Proc. Cambridge Philos. Soc. 136(3) (2004), 513–524]. In particular, the preservation of word-hyperbolicity under taking free products is considered. Under mild conditions on the semigroups involved, satisfied, for example, by monoids or regular semigroups, we prove that the semigroup free product of two word-hyperbolic semigroups is again word-hyperbolic. Analogously, with a mild condition on the uniqueness of representation for the identity element, satisfied, for example, by groups, we prove that the monoid free product of two word-hyperbolic monoids is word-hyperbolic. The methods are language-theoretically general, and apply equally well to semigroups, monoids or groups with a $\mathbf {C}$-multiplication table, where $\mathbf {C}$ is any reversal-closed super-$\operatorname {\mathrm {AFL}}$. In particular, we deduce that the free product of two groups with $\mathbf {ET0L}$ with respect to indexed multiplication tables again has an $\mathbf {ET0L}$ with respect to an indexed multiplication table.

We present a metric condition $\TTMetric$ which describes the geometry of classical small cancellation groups and applies also to other known classes of groups such as two-dimensional Artin groups. We prove that presentations satisfying condition $\TTMetric$ are diagrammatically reducible in the sense of Sieradski and Gersten. In particular, we deduce that the standard presentation of an Artin group is aspherical if and only if it is diagrammatically reducible. We show that, under some extra hypotheses, $\TTMetric$-groups have quadratic Dehn functions and solvable conjugacy problem. In the spirit of Greendlinger's lemma, we prove that if a presentation P = 〈X| R〉 of group G satisfies conditions $\TTMetric -C'(\frac {1}{2})$, the length of any nontrivial word in the free group generated by X representing the trivial element in G is at least that of the shortest relator. We also introduce a strict metric condition $\TTMetricStrict$, which implies hyperbolicity.

Subshifts with property $(A)$ are constructed from a class of directed graphs. As special cases the Markov–Dyck shifts are shown to have property $(A)$. The semigroups that are associated to ${\mathcal{R}}$-graph shifts with Property $(A)$ are determined.

Let $T_{n}(\mathbb{F})$ be the semigroup of all upper triangular $n\times n$ matrices over a field $\mathbb{F}$. Let $UT_{n}(\mathbb{F})$ and $UT_{n}^{\pm 1}(\mathbb{F})$ be subsemigroups of $T_{n}(\mathbb{F})$, respectively, having $0$s and/or $1$s on the main diagonal and $0$s and/or $\pm 1$s on the main diagonal. We give some sufficient conditions under which an involution semigroup is nonfinitely based. As an application, we show that $UT_{2}(\mathbb{F}),UT_{2}^{\pm 1}(\mathbb{F})$ and $T_{2}(\mathbb{F})$ as involution semigroups under the skew transposition are nonfinitely based for any field $\mathbb{F}$.

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.

This paper studies the growths of endomorphisms of finitely generated semigroups. The growth is a certain dynamical characteristic describing how iterations of the endomorphism ‘stretch’ balls in the Cayley graph of the semigroup. We make a detailed study of the relation of the growth of an endomorphism of a finitely generated semigroup and the growth of the restrictions of the endomorphism to finitely generated invariant subsemigroups. We also study the possible values endomorphism growths can attain. We show the role of linear algebra in calculating the growths of endomorphisms of homogeneous semigroups. Proofs are a mixture of syntactic algebraic rewriting techniques and analytical tricks. We state various problems and suggestions for future research.

A monoid S is said to be right coherent if every finitely generated subact of every finitely presented right S-act is finitely presented. Left coherency is defined dually and S is coherent if it is both right and left coherent. These notions are analogous to those for a ring R (where, of course, S-acts are replaced by R-modules). Choo et al. have shown that free rings are coherent. In this paper we prove that, correspondingly, any free monoid is coherent, thus answering a question posed by Gould in 1992.

We show that every finitely generated algebra that is a finitely generated module over a finitely generated commutative subalgebra is an automaton algebra in the sense of Ufnarovskii.

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.

This paper revisits the solution of the word problem for ${\it\omega}$-terms interpreted over finite aperiodic semigroups, obtained by J. McCammond. The original proof of correctness of McCammond’s algorithm, based on normal forms for such terms, uses McCammond’s solution of the word problem for certain Burnside semigroups. In this paper, we establish a new, simpler, correctness proof of McCammond’s algorithm, based on properties of certain regular languages associated with the normal forms. This method leads to new applications.

Two semigroups are distinct if they are neither isomorphic nor anti-isomorphic. Although there exist $15\,973$ pairwise distinct semigroups of order six, only four are known to be non-finitely based. In the present article, the finite basis property of the other $15\,969$ distinct semigroups of order six is verified. Since all semigroups of order five or less are finitely based, the four known non-finitely based semigroups of order six are the only examples of minimal order.

For a finite Clifford inverse algebra $A$, with natural order meet-semilattice ${Y}_{A} $ and group of units ${G}_{A} $, we show that the inverse monoid obtained as the semidirect product ${ Y}_{A}^{1} {\mathop{\ast }\nolimits}_{\rho } {G}_{A} $ has a log-polynomial free spectrum whenever $\rho $ is a term-expressible left action of ${G}_{A} $ on ${Y}_{A} $ and all subgroups of $A$ are nilpotent. This yields a number of examples of finite inverse monoids satisfying the Seif conjecture on finite monoids whose free spectra are not doubly exponential.