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Nik Ruskuc

Publications and source records attributed to Nik Ruskuc.

At least 19 recordsLinked to original sources
arXiv2025

Maximal subgroups of free projection- and idempotent-generated semigroups with applications to partition monoids

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 $r\leq n-2$ is (isomorphic to) the symmetric group $S_r$. In $IG(E(P_n))$, the corresponding subgroup is the direct product $Z \times S_r$. The appearance of the infinite cyclic group $Z$ is explained by a connection to a certain twisted partition monoid $P_n^\Phi$, which has the same biordered set as $P_n$.

math.GR↗
arXiv2024

On Finite Presentability of Subsemigroups of the Monogenic Free Inverse Semigroup

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.

math.GR↗
arXiv2024

Generators and presentations of inverse subsemigroups of the monogenic free inverse semigroup

It was proved by Oliveira and Silva (2005) that every finitely generated inverse subsemigroup of the monogenic free inverse semigroup $FI_1$ is finitely presented. The present paper continues this development, and gives generating sets and presentations for general (i.e. not necessarily finitely generated) inverse subsemigroups of $FI_1$. For an inverse semigroup $S$ and an inverse subsemigroup $T$ of $S$, we say $S$ is finitely generated modulo $T$ if there is a finite set $A$ such that $S = \langle T, A \rangle$. Likewise, we say that $S$ is finitely presented modulo $T $ if $S$ can be defined by a presentation of the form $\text{Inv}\langle X, Y \mid R, Q\rangle$, where $\text{Inv}\langle X\mid R\rangle$ is a presentation for $T$ and $Y$ and $Q$ are finite. We show that every inverse subsemigroup $S$ of $FI_1$ is finitely generated modulo its semilattice of idempotents $E(S)$. By way of contrast, we show that when $S\neq E(S)$, it can never be finitely presented modulo $E(S)$. However, in the process we establish some nice (albeit infinite) presentations for $S$ modulo $E(S)$.

math.GR↗
arXiv2023

Coherency properties for monoids of transformations and partitions

A monoid $S$ is right coherent if every finitely generated subact of every finitely presented right $S$-act itself has a finite presentation; it is weakly right coherent if every finitely generated right ideal of $S$ has a finite presentation. We show that full and partial transformation monoids, symmetric inverse monoids and partition monoids over an infinite set are all weakly right coherent, but that none of them is right coherent. Left coherency and weak left coherency are defined dually, and the corresponding results hold for these properties. In order to prove the non-coherency results, we give a presentation of an inverse semigroup which does not embed into any left or right coherent monoid.

math.RA↗
arXiv2023

On the number of subdirect products involving semigroups of integers and natural numbers

We extend a recent result that for the (additive) semigroup of positive integers $\mathbb{N}$, there are continuum many subdirect products of $\mathbb{N} \times \mathbb{N}$ up to isomorphism. We prove that for $U,V$ each one of $\mathbb{Z}$ (the group of integers), $\mathbb{N}_{0}$ (the monoid of non-negative integers), or $\mathbb{N}$, we prove that $U \times V$ has continuum many (semigroup) subdirect products up to isomorphism.

math.GR↗
arXiv2023

On the diameter of semigroups of transformations and partitions

For a semigroup $S$ whose universal right congruence is finitely generated (or, equivalently, a semigroup satisfying the homological finiteness property of being type right-$FP_1$), the right diameter of $S$ is a parameter that expresses how `far apart' elements of $S$ can be from each other, in a certain sense. To be more precise, for each finite generating set $U$ for the universal right congruence on $S,$ we have a metric space $(S,d_U)$ where $d_U(a,b)$ is the minimum length of derivations for $(a,b)$ as a consequence of pairs in $U$; the right diameter of $S$ with respect to $U$ is the diameter of this metric space. The right diameter of $S$ is then the minimum of the set of all right diameters with respect to finite generating sets. We investigate whether various natural infinite semigroups of transformations and partitions have a finitely generated universal right/left congruence, and for those that do, we determine their right/left diameter. Among other results, for an arbitrary infinite set $X$ we prove the following. Each of the monoids of all binary relations on $X,$ of all partial transformations on $X,$ and of all full transformations on $X,$ as well as the partition and partial Brauer monoids on $X,$ have right diameter 1 and left diameter 1. The symmetric inverse monoid on $X$ has right diameter 2 and left diameter 2. The monoid of all injective mappings on $X$ has right diameter 4, and its minimal ideal (called the Baer-Levi semigroup on $X$) has right diameter 3, but neither of these two semigroups has a finitely generated universal left congruence. On the other hand, the semigroup of all surjective mappings on $X$ has left diameter 4, and its minimal ideal has left diameter 2, but neither of these semigroups has a finitely generated universal right congruence.

math.GR↗
arXiv2022

On the number of countable subdirect powers of finite commutative semigroups

In 1981/82, Hickin \& Plotkin and McKenzie both proved that a finite group has only countably many non-isomorphic subdirect powers if and only if it is abelian. In this paper, we prove that a finite commutative semigroup has only countably many non-isomorphic countable subdirect powers if and only if it is either a finite abelian group or a null semigroup.

math.GR↗
arXiv2022

On minimal ideals in pseudo-finite semigroups

A semigroup $S$ is said to be right pseudo-finite if the universal right congruence can be generated by a finite set $U\subseteq S\times S$, and there is a bound on the length of derivations for an arbitrary pair $(s,t)\in S\times S$ as a consequence of those in $U$. This article explores the existence and nature of a minimal ideal in a right pseudo-finite semigroup. Continuing the theme started in an earlier work by Dandan et al., we show that in several natural classes of monoids, right pseudo-finiteness implies the existence of a completely simple minimal ideal. This is the case for orthodox monoids, completely regular monoids and right reversible monoids, which include all commutative monoids. We also show that certain other conditions imply the existence of a minimal ideal, which need not be completely simple; notably, this is the case for semigroups in which one of the Green's pre-orders $\leq_{\mathcal{L}}$ or $\leq_{\mathcal{J}}$ is left compatible with multiplication. Finally, we establish a number of examples of pseudo-finite monoids without a minimal ideal. We develop an explicit construction that yields such examples with additional desired properties, for instance, regularity or $\mathcal{J}$-triviality.

math.GR↗
arXiv2021

Properties of congruences of twisted partition monoids and their lattices

We build on the recent characterisation of congruences on the infinite twisted partition monoids $\mathcal{P}_{n}^Φ$ and their finite $d$-twisted homomorphic images $\mathcal{P}_{n,d}^Φ$, and investigate their algebraic and order-theoretic properties. We prove that each congruence of $\mathcal{P}_{n}^Φ$ is (finitely) generated by at most $\lceil\frac{5n}2\rceil$ pairs, and we characterise the principal ones. We also prove that the congruence lattice $\textsf{Cong}(\mathcal{P}_{n}^Φ)$ is not modular (or distributive); it has no infinite ascending chains, but it does have infinite descending chains and infinite antichains. By way of contrast, the lattice $\textsf{Cong}(\mathcal{P}_{n,d}^Φ)$ is modular but still not distributive for $d>0$, while $\textsf{Cong}(\mathcal{P}_{n,0}^Φ)$ is distributive. We also calculate the number of congruences of $\mathcal{P}_{n,d}^Φ$, showing that the array $\big(|\textsf{Cong}(\mathcal{P}_{n,d}^Φ)|\big)_{n,d\geq 0}$ has a rational generating function, and that for a fixed $n$ or $d$, $|\textsf{Cong}(\mathcal{P}_{n,d}^Φ)|$ is a polynomial in $d$ or $n\geq 4$, respectively.

math.RA↗
arXiv2021

Classification of congruences of twisted partition monoids

The twisted partition monoid $\mathcal{P}_n^Φ$ is an infinite monoid obtained from the classical finite partition monoid $\mathcal{P}_n$ by taking into account the number of floating components when multiplying partitions. The main result of this paper is a complete description of the congruences on $\mathcal{P}_n^Φ$. The succinct encoding of a congruence, which we call a C-pair, consists of a sequence of $n+1$ congruences on the additive monoid $\mathbb{N}$ of natural numbers and a certain $(n+1)\times\mathbb{N}$ matrix. We also give a description of the inclusion ordering of congruences in terms of a lexicographic-like ordering on C-pairs. This is then used to classify congruences on the finite $d$-twisted partition monoids $\mathcal{P}_{n,d}^Φ$, which are obtained by factoring out from $\mathcal{P}_n^Φ$ the ideal of all partitions with more than $d$ floating components. Further applications of our results, elucidating the structure and properties of the congruence lattices of the ($d$-)twisted partition monoids, will be the subject of a future article.

math.RA↗
arXiv2021

On separability finiteness conditions in semigroups

Taking residual finiteness as a starting point, we consider three related finiteness properties: weak subsemigroup separability, strong subsemigroup separability and complete separability. We investigate whether each of these properties is inherited by Schützenberger groups. The main result of this paper states that for a finitely generated commutative semigroup $S$, these three separability conditions coincide and are equivalent to every $\mathcal{H}$-class of $S$ being finite. We also provide examples to show that these properties in general differ for commutative semigroups and finitely generated semigroups. For a semigroup with finitely many $\mathcal{H}$-classes, we investigate whether it has one of these properties if and only if all its Schützenberger groups have the property.

math.GR↗
arXiv2021

Congruences on infinite partition and partial Brauer monoids

We give a complete description of the congruences on the partition monoid $P_X$ and the partial Brauer monoid $PB_X$, where $X$ is an arbitrary infinite set, and also of the lattices formed by all such congruences. Our results complement those from a recent article of East, Mitchell, Ruskuc and Torpey, which deals with the finite case. As a consequence of our classification result, we show that the congruence lattices of $P_X$ and $PB_X$ are isomorphic to each other, and are distributive and well quasi-ordered. We also calculate the smallest number of pairs of partitions required to generate any congruence; when this number is infinite, it depends on the cofinality of certain limit cardinals.

math.GR↗
arXiv2021

On separability properties in direct products of semigroups

We investigate four finiteness conditions related to residual finiteness: complete separability, strong subsemigroup separability, weak subsemigroup separability and monogenic subsemigroup separability. For each of these properties we examine under which conditions the property is preserved under direct products. We also consider if any of the properties are inherited by the factors in a direct product. We give necessary and sufficient conditions for finite semigroups to preserve the properties of strong subsemigroup separability and monogenic subsemigroup separability in a direct product.

math.RA↗
arXiv2021

On groups of units of special and one-relator inverse monoids

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.

math.GR↗
arXiv2020

Atomicity and well quasi-order for consecutive orderings on words and permutations

Algorithmic decidability is established for two order-theoretic properties of downward closed subsets defined by finitely many obstructions in two infinite posets. The properties under consideration are: (a) being atomic, i.e. not being decomposable as a union of two downward closed proper subsets, or, equivalently, satisfying the joint embedding property; and (b) being well quasi-ordered. The two posets are: (1) words over a finite alphabet under the consecutive subword ordering; and (2) finite permutations under the consecutive subpermutation ordering. Underpinning the four results are characterisations of atomicity and well quasi-order for the subpath ordering on paths of a finite directed graph.

math.CO↗
arXiv2020

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.

math.GR↗
arXiv2020

Congruence lattices of ideals in categories and (partial) semigroups

This paper presents a unified framework for determining the congruences on a number of monoids and categories of transformations, diagrams, matrices and braids, and on all their ideals. The key theoretical advances present an iterative process of stacking certain normal subgroup lattices on top of each other to successively build congruence lattices of a chain of ideals. This is applied to several specific categories of: transformations; order/orientation preserving/reversing transformations; partitions; planar/annular partitions; Brauer, Temperley--Lieb and Jones partitions; linear and projective linear transformations; and partial braids. Special considerations are needed for certain small ideals, and technically more intricate theoretical underpinnings for the linear and partial braid categories.

math.GR↗