A monoid presentation is called special if the right-hand side of each defining relation is equal to 1. We prove results which relate the two-sided homological finiteness properties of a monoid defined by a special presentation with those of its group of units. Specifically we show that the monoid enjoys the homological finiteness property bi- if its group of units is of type . We also obtain results which relate the Hochschild cohomological dimension of the monoid to the cohomological dimension of its group of units. In particular we show that the Hochschild cohomological dimension of the monoid is bounded above by the maximum of $2$ and the cohomological dimension of its group of units. We apply these results to prove a Lyndon's Identity type theorem for the two-sided homology of one-relator monoids of the form $\langle A \mid r=1 angle $ . In particular, we show that all such monoids are of type bi-. Moreover, we show that if r is not a proper power then the one-relator monoid has Hochschild cohomological dimension at most $2$ , while if r is a proper power then it has infinite Hochschild cohomological dimension. For any nonspecial one-relator monoid M with defining relation $u=v$ we show that if there is no nonempty word w such that $u,v \in A<^>*w \cap w A<^>*$ then M is of type bi- $\mathrm {FP}_\infty $ and has Hochschild cohomological dimension at most $2$ .

This paper investigates the maximal subgroups of a free projection-generated regular *-semigroup PG(P) over a projection algebra P, and their relationship to the maximal subgroups of the free idempotent-generated semigroup IG(E) over the corresponding biordered set E = E(P). In the first part of the paper we obtain a number of general presentations by generators and defining relations, in each case reflecting salient combinatorial/topological properties of the groups. In the second part we apply these to explicitly compute the groups when P = P(P_n) and E = E(P_n) arise from the partition monoid P_n. Specifically, we show that the maximal subgroup of PG(P(P_n)) corresponding to a projection of rank r≤ n-2 is (isomorphic to) the symmetric group S_r. In IG(E(P_n)), the corresponding subgroup is the direct product Z × S_r. The appearance of the infinite cyclic group Z is explained by a connection to a certain twisted partition monoid P_n^Φ, which has the same biordered set as P_n.

We study the class of monoids that arise as the submonoid of right units of finitely presented special inverse monoids. Classical results of Makanin show that the monoids of right units in finitely presented special monoids decompose as a free product of the group of units and a finite rank free monoid. Gray and Ruškuc (2024) gave the first example of a finitely presented special inverse monoid whose submonoid of right units does not admit such a decomposition, which left open the question of determining the structure of such monoids. In the first part of this paper we prove a general result which shows that the only instances where the right units of a finitely presented special inverse monoid can admit such a free product decomposition are when their group of units is finitely presented. This implies that the typical behaviour of the right units is not to admit such a free product decomposition. In showing this, we establish some general results about finite generation and presentability of subgroups of special inverse monoids. In particular, we give an exact characterisation of when an arbitrary subgroup is finitely generated in terms of connectedness properties of unions of its cosets in its R-class, and also a characterisation of when an arbitrary subgroup is finitely presented. We also give a sufficient condition for finite generation and presentability of an arbitrary subgroup given in terms of a geometric finiteness property called boundary width. As a consequence, we show that the classes of monoids of right units of finitely presented special inverse monoids and prefix monoids of finitely presented groups are independent, in the sense that neither of them is contained in the other. In the second part of the paper, we show that every finitely generated submonoid of a finitely RC-presented monoid is isomorphic to a submonoid N of a finitely presented special inverse monoid M such that N is a submonoid of the right units of M, and N contains the group of units of M. This result generalises and extends the classification of groups of units of finitely presented special inverse monoids recently obtained by Gray and Kambites (2025). From this, we derive a number of surprising properties of RC-presentations for right cancellative monoids contrasting the classical theory of monoid presentations.

We prove that any finite graph of finitely generated free groups admits a finite complete rewriting system after possibly taking a free product with a free group of rank two. As a corollary we obtain that any HNN-extension of a finitely generated free group over a finitely generated subgroup admits a finite complete rewriting system. We then use this result, and other tools, to give partial solutions to several fundamental open problems about finite complete rewriting systems for hyperbolic, one-relator, fully residually free, and three-manifold groups. In particular we prove that if G = ⟨𝔽 , t | t^-1ft = ψ(f), ∀ f∈𝔽⟩ is the mapping torus of an injective endomorphism ψ of a free group 𝔽 (of possibly infinite rank) then every finitely generated subgroup of G admits a finite complete rewriting system. It follows that any finitely generated virtually free-by-cyclic group, and any finitely generated subgroup of such a group, admits a finite complete rewriting system. We apply this to show that every finitely generated subgroup of a locally quasi-convex hyperbolic and virtually compact special group admits a finite complete rewriting system. This includes all one-relator groups with torsion (and all their finitely generated subgroups) and all hyperbolic fully residually free groups. Moreover, we show there is an algorithm that computes a finite complete rewriting system for any such group, given a presentation for its containing group and a finite list of generators for the subgroup. We also prove that for every compact three-manifold M, the group π_1(M) ∗ℤ admits a finite complete rewriting system. Furthermore, we show that the fundamental group of any compact three-manifold is autostackable and thus has a rational cross section and admits a bounded regular convergent prefix-rewriting system.

We study the word problem in two-generator one-relator inverse monoids, proving both decidability and undecidability results. We give the first example of a two-generator one-relator inverse monoid of the form Inv⟨ a,b | w=1 ⟩ with an undecidable word problem. Furthermore our examples can be chosen to be E-unitary. To prove this result we exploit the connection between the word problem for one-relator inverse monoids and the submonoid membership problem for one-relator groups established in work of Ivanov, Margolis, and Meakin (2001). We also give the first example of a two-generator one-relator group, with a reduced word defining relator, with an undecidable prefix membership problem. We investigate the problem of classifying the two-generator one-relator inverse mononids with decidable word problem. For this we prove a general result classifying when a free-by-cyclic group with a polynomially growing monodromy has decidable submonoid membership problem. Specifically we prove that a free-by-cyclic group with a polynomially growing monodromy has decidable submonoid membership problem if and only if it has decidable rational subset membership problem if and only if the defining automorphism has finite order in the outer automorphism group. We then apply that result to give a sufficient condition for a two-generator one-relator inverse monoid to have decidable word problem in the case its maximal group image is free-by-cyclic of polynomial growth, and we show that that condition can be algorithmically checked. Conversely, we show that when this condition is not satisfied then the two-generator one-relator inverse monoid can be lifted to an example with undecidable word problem and with the same maximal group image (up to taking a free product with the infinite cyclic group).

A twisting of a monoid S is a map I' : S x S-* N satisfying the identity I'(a, b) + I'(ab, c) = I'(a, bc) + I'(b, c). Together with an additive commutative monoid M, and a fixed q E M, this gives rise a so-called twisted product M xq Phi S, which has underlying set M x Sand multiplication (i, a)(j, b) = (i + j + I'(a, b)q, ab). This construction has appeared in the special cases where M is Nor Z under addition, S is a diagram monoid (e.g. partition, Brauer or Temperley-Lieb), and I' counts floating components in concatenated diagrams. In this paper we identify a special kind of 'tight' twisting, and give a thorough structural description of the resulting twisted products. This involves characterising Green's relations, (von Neumann) regular elements, idempotents, biordered sets, maximal subgroups, Sch & uuml;tzenberger groups, and more. We also consider a number of examples, including several apparently new ones, which take as their starting point certain generalisations of Sylvester's rank inequality from linear algebra. (c) 2025 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY license (http:// creativecommons.org/licenses/by/4.0/).

Motivated by its applications to the word problem for one-relator inverse monoids, via results of Ivanov, Margolis, and Meakin (2001), we prove several decidability and undecidability results about the submonoid membership problem in one-relator, surface and hyperbolic groups. We prove a general result that gives sufficient conditions under which membership in submonoids of certain free-by-cyclic one-relator groups is decidable. We apply this result to show that membership in various families of submonoids of surface groups is decidable. We study graded submonoids of one-relator groups, proving results that give conditions under which such a submonoid has decidable membership with linear distortion. These results are then applied to improve on a result of Margolis, Meakin and Šunik by showing the prefix monoid of a surface group has linear distortion. Our results significantly extend previously known positive results proving decidability of membership in (non group) submonoids of surface groups. We also apply our general results, and other techniques, to show that the Magnus submonoid membership problem is decidable in several families of one-relator groups including: surface groups, Baumslag–Solitar groups, and in certain free-by-cyclic one-relator groups. We also prove results about the related positivity problem (also called the quasi-Magnus problem) which asks whether membership in the submonoid generate by A is decidable. We resolve a problem posed by McCammond and Meakin in 2006 by showing that there is a hyperbolic group G generated by A such that it is undecidable whether an element can be represented by a positive word on the generators A. We show in addition this hyperbolic group can be chosen to be residually finite. We do this by giving a new general method for constructing finitely presented groups with undecidable positivity problem.

Motivated by the open problem of whether all one-relator groups have decidable Diophantine problem, in this paper we prove a collection of undecidability results about the Diophantine problem for several families of groups that are close to one-relator groups in various ways. We prove that there is a generalised Baumslag–Solitar group with an undecidable Diophantine problem. Using our example we show there is a group with an undecidable Diophantine problem that is quasi-isometric to a one-relator group. Also, we prove that there is a one-relator product of cyclic groups with an undecidable Diophantine problem. In addition, we show that there there is a one-relator group G, with a single fixed finite rank free subgroup H, such that the Diophantine problem for G with H-constraints is undecidable. The related open question of whether there is a free-by-cyclic group with undecidable Diophantine problem is also discussed, and we prove that there is a free-by-free group of the form F_3 ⋊ F_2 with an undecidable Diophantine problem.

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 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 non-empty. 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.

We study the maximal subgroups (also known as group \mathcal{H} -classes) of finitely presented special inverse monoids. We show that the maximal subgroups which can arise in such monoids are exactly the recursively presented groups, and moreover every such maximal subgroup can also arise in the E -unitary case. We also prove that the possible groups of units are exactly the finitely generated recursively presented groups; this improves upon a result of, and answers a question of, the first author and Ruškuc. These results give the first significant insight into the maximal subgroups of such monoids beyond the group of units, and the results together demonstrate that (perhaps surprisingly) it is possible for the subgroup structure to have a complexity which significantly exceeds that of the group of units. We also observe that a finitely presented special inverse monoid (even an E -unitary one) may have infinitely many pairwise non-isomorphic maximal subgroups; this contrasts sharply with the case of (non-inverse) special monoids, where Malheiro showed that all idempotents lie in the \mathcal{D} -class of 1 , from which it follows that all maximal subgroups are isomorphic.

We study several natural decision problems in braid groups and Artin groups. We classify the Artin groups with decidable submonoid membership problem in terms of the nonexistence of certain forbidden-induced subgraphs of the defining graph. Furthermore, we also classify the Artin groups for which the following problems are decidable: the rational subset membership problem, semigroup intersection problem, and the fixed-target submonoid membership problem. In the case of braid groups, our results show that the submonoid membership problem, and each and every one of these problems, is decidable in the braid group B-n if and only if n <= 3, which answers an open problem of Potapov (2013). Our results also generalize and extend results of Lohrey and Steinberg (2008) who classified right-angled Artin groups with decidable submonoid (and rational subset) membership problem.

The purpose of this paper is to introduce a new family of semigroups the free projection-generated regular *semigroups and initiate their systematic study. Such a semigroup PG(P) is constructed from a projection algebra P, using the recent groupoid approach to regular *-semigroups. The assignment P bar right arrow PG(P)is a left adjoint to the forgetful functor that maps a regular *-semigroup S to its projection algebra P(S). In fact, the category of projection algebras is coreflective in the category of regular (*)-semigroups. The algebra P(S) uniquely determines the biordered structure of the idempotents E(S), up to isomorphism, and this leads to a category equivalence between projection algebras and regular (*)-biordered sets. As a consequence, PG(P) can be viewed as a quotient of the classical free idempotent-generated (regular) The purpose of this paper is to introduce a new family of semigroups the free projection-generated regular (*)semigroups and initiate their systematic study. Such a semigroup PG(P) is constructed from a projection algebra P, using the recent groupoid approach to regular (*)-semigroups. The assignment P bar right arrow PG(P)is a left adjoint to the forgetful functor that maps a regular *-semigroup S to its projection algebra P(S). In fact, the category of projection algebras is coreflective in the category of regular (*)-semigroups. The algebra P(S) uniquely determines the biordered structure of the idempotents E(S), up to isomorphism, and this leads to a category equivalence between projection algebras and regular (*)-biordered sets. As a consequence, PG(P) can be viewed as a quotient of the classical free idempotent-generated (regular) semigroups IG(E) and RIG(E), where E = E(PG(P)); this is witnessed by a number of presentations in terms of genera-tors and defining relations. The semigroup PG(P) can also be interpreted topologically, through a natural link to the funda-mental groupoid of a simplicial complex explicitly constructed from P. The above theory is illustrated on a number of ex-amples. In one direction, the free construction applied to the projection algebras of adjacency semigroups yields a new fam-ily of graph-based path semigroups. In another, it turns out that, remarkably, the Temperley-Lieb monoid TCnis the free regular (*)-semigroup over its own projection algebra P(TCn). (c) 2025 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY license (http:// creativecommons.org/licenses/by/4.0/).

We continue the study of the structure of general subgroups (in particular maximal subgroups, also known as group $$\mathcal {H}$$ H -classes) of special inverse monoids. Recent research of the authors has established that these can be quite wild, but in this paper we show that if we restrict to special inverse monoids which are E-unitary (or have a weaker property we call $$\mathcal {R}_1$$ R 1 -injectivity), the maximal subgroups are strongly governed by the group of units. In particular, every maximal subgroup has a finite index subgroup which embeds in the group of units. We give a construction to show that every finite group can arise as a maximal subgroup in an $$\mathcal {R}_1$$ R 1 -injective special inverse monoid with trivial group of units. It remains open whether every combination of a group G and finite index subgroup H can arise as maximal subgroup and group of units.

We show how topological methods developed in a previous article can be applied to prove new results about topological and homological finiteness properties of monoids. A monoid presentation is called special if the right-hand side of each relation is equal to 1 . We prove results which relate the finiteness properties of a monoid defined by a special presentation with those of its group of units. Specifically we show that the monoid inherits the finiteness properties F n and FP n from its group of units. We also obtain results which relate the geometric and cohomological dimensions of such a monoid to those of its group of units. We apply these results to prove a Lyndon's Identity Theorem for one -relator monoids of the form h A j r D 1 i . In particular, we show that all such monoids are of type F 1 (and FP 1 ), and that when r is not a proper power, then the monoid has geometric and cohomological dimension at most 2 . The first of these results, resolves an important case of a question of Kobayashi from 2000 on homological finiteness properties of one -relator monoids. We also show how our topological approach can be used to prove results about the closure properties of various homological and topological finiteness properties for amalgamated free products and HNN-extensions of monoids. To prove these results we introduce new methods for constructing equivariant classifying spaces for monoids, as well as developing a Bass-Serre theory for free constructions of monoids.

AbstractA 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.

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.

We prove that the class of finitely presented inverse monoids whose Schützenberger graphs are quasi-isometric to trees has a uniformly solvable word problem, furthermore, the languages of their Schützenberger automata are context-free. On the other hand, we show that there is a finitely presented inverse monoid with hyperbolic Schützenberger graphs and an unsolvable word problem.

We initiate the study of higher dimensional topological finiteness properties of monoids. This is done by developing the theory of monoids acting on CW complexes. For this we establish the foundations of $M$-equivariant homotopy theory where $M$ is a discrete monoid. For projective $M$-CW complexes we prove several fundamental results such as the homotopy extension and lifting property, which we use to prove the $M$-equivariant Whitehead theorems. We define a left equivariant classifying space as a contractible projective $M$-CW complex. We prove that such a space is unique up to $M$-homotopy equivalence and give a canonical model for such a space via the nerve of the right Cayley graph category of the monoid. The topological finiteness conditions left-$\mathrm{F}_n$ and left geometric dimension are then defined for monoids in terms of existence of a left equivariant classifying space satisfying appropriate finiteness properties. We also introduce the bilateral notion of $M$-equivariant classifying space, proving uniqueness and giving a canonical model via the nerve of the two-sided Cayley graph category, and we define the associated finiteness properties bi-$\mathrm{F}_n$ and geometric dimension. We explore the connections between all of the these topological finiteness properties and several well-studied homological finiteness properties of monoids which are important in the theory of string rewriting systems, including $\mathrm{FP}_n$, cohomological dimension, and Hochschild cohomological dimension. We also develop the corresponding theory of $M$-equivariant collapsing schemes (that is, $M$-equivariant discrete Morse theory), and among other things apply it to give topological proofs of results of Anick, Squier and Kobayashi that monoids which admit presentations by complete rewriting systems are left-, right- and bi-$\mathrm{FP}_\infty$.

For every one-relator monoid $M = \langle A \mid u=v \rangle$ with $u, v \in A^*$ we construct a contractible $M$-CW complex and use it to build a projective resolution of the trivial module which is finitely generated in all dimensions. This proves that all one-relator monoids are of type ${\rm FP}_\infty$, answering positively a problem posed by Kobayashi in 2000. We also apply our results to classify the one-relator monoids of cohomological dimension at most $2$, and to describe the relation module, in the sense of Ivanov, of a torsion-free one-relator monoid presentation as an explicitly given principal left ideal of the monoid ring. In addition, we prove the topological analogues of these results by showing that all one-relator monoids satisfy the topological finiteness property ${\rm F}_\infty$, and classifying the one-relator moniods with geometric dimension at most $2$. These results give a natural monoid analogue of Lyndon's Identity Theorem for one-relator groups.

The workshop "Homogeneous Structures, A Workshop in Honour of Norbert Sauer's 70th Birthday" took place at the Banff International Research Station from November 8th to November 13th, 2015. The purpose of the following note is to gather the list of open problems that were posed during the problem sessions. Contributions appear in alphabetical order, according to the name of their author.