Tied monoids

We construct certain monoids, called tied monoids. These monoids result to be semidirect products finitely presented and commonly built from braid groups and their relatives acting on monoids of set partitions. The nature of our monoids indicate that they should give origin to new knot algebras; indeed, our tied monoids include the tied braid monoid and the tied singular braid monoid, which were used, respectively, to construct new polynomial invariants for classical links and singular links. Consequently, we provide a mechanism to attach an algebra to each tied monoid; this mechanism not only captures known generalizations of the bt-algebra, but also produces possible new knot algebras. To build the tied monoids it is necessary to have presentations of set partition monoids of types A, B and D, among others. For type A we use a presentation due to FitzGerald and for the other type it was necessary to built them.

     

On strongly (<Emphasis Type="Italic">P</Emphasis>)-cyclic acts

By a regular act we mean an act such that all its cyclic subacts are projective. In this paper we introduce strong (P)-cyclic property of acts over monoids which is an extension of regularity and give a classification of monoids by this property of their right (Rees factor) acts.     

SEPARABILITY CONDITIONS IN ACTS OVER MONOIDS

We discuss residual finiteness and several related separability conditions for the class of monoid acts, namely weak subact separability, strong subact separability and complete separability. For each of these four separability conditions, we investigate which monoids have the property that all their (finitely generated) acts satisfy the condition. In particular, we prove that: all acts over a finite monoid are completely separable (and hence satisfy the other three separability conditions); all finitely generated acts over a finitely generated commutative monoid are residually finite and strongly subact separable (and hence weakly subact separable); all acts over a commutative idempotent monoid are residually finite and strongly subact separable; and all acts over a Clifford monoid are strongly subact separable.

     

Constructing finitely presented monoids which have no finite complete presentation

Squier (1987) showed that if a monoid is defined by a finite complete rewriting system, then it satisfies the homological finiteness condition FP₃, and using this fact he gave monoids (groups) which have solvable word problems but cannot be presented by finite complete systems. In the present paper we show that a monoid cannot have a finite complete presentation if it contains certain special elements. This observation enables us to construct monoids without finite complete presentation in a direct and elementary way. We give a finitely presented monoid which has (1) a word problem solvable in linear time and (2) linear growth but (3) no finite complete presentation. We also give a finitely presented monoid which has (1) a word problem solvable in linear time, (2) finite derivation type in the sense of Squier and (3) the property FP_∞, but (4) no finite complete presentation.     

Strongly Flat and Condition (P) Covers of Acts Over Monoids

In this article we characterize monoids over which every right S-act has a strongly flat (condition (P)) cover. Similar to the perfect monoids, such monoids are characterized by condition (A) and having strongly flat (condition (P)) cover for each cyclic right S-act. We also give a new characterization for perfect monoids as monoids over which every strongly flat right S-act has a projective cover.     

Regular S-acts with primitive normal and antiadditive theories

In this work, we investigate the commutative monoids over which the axiomatizable class of regular S-acts is primitive normal and antiadditive. We prove that the primitive normality of an axiomatizable class of regular S-acts over the commutative monoid S is equivalent to the antiadditivity of this class and it is equivalent to the linearity of the order of a semigroup R such that an S-act sR is a maximal (under the inclusion) regular subact of the S-act sS.     

Principal ideals of finitely generated commutative monoids

We study the semigroups isomorphic to principal ideals of finitely generated commutative monoids. We define the concept of finite presentation for this kind of semigroups. Furthermore, we show how to obtain information on these semigroups from their presentations.     

On Quiver Grassmannians and Orbit Closures for Gen-Finite Modules

We show that endomorphism rings of cogenerators in the module category of a finite-dimensional algebra A admit a canonical tilting module, whose tilted algebra B is related to A by a recollement. Let M be a gen-finite A-module, meaning there are only finitely many indecomposable modules generated by M. Using the canonical tilts of endomorphism algebras of suitable cogenerators associated to M, and the resulting recollements, we construct desingularisations of the orbit closure and quiver Grassmannians of M, thus generalising all results from previous work of Crawley-Boevey and the second author in 2017. We provide dual versions of the key results, in order to also treat cogen-finite modules.

     

Separable algebras over infinite fields are 2-generated and finitely presented

We prove that every separable algebra over an infinite field F admits a presentation with 2 generators and finitely many relations. In particular, this is true for finite direct sums of matrix algebras over F and for group algebras FG, where G is a finite group such that the order of G is invertible in F. We illustrate the usefulness of such presentations by using them to find a polynomial criterion to decide when 2 ordered pairs of 2 × 2 matrices (A, B) and (A′, B′) with entries in a commutative ring R are automorphically conjugate over the matrix algebra M ₂(R), under an additional assumption that both pairs generate M ₂(R) as an R-algebra.     

Embedding Free Burnside Groups in Finitely Presented Groups

We construct an embedding of a free Burnside group B(m,n) of odd exponent n > 2⁴⁸ and rank m >1 in a finitely presented group with some special properties. The main application of this embedding is an easy construction of finitely presented nonamenable groups without noncyclic free subgroups (which provides a new finitely presented counterexample to the von Neumann problem on amenable groups). As another application, we construct weakly finitely presented groups of odd exponent n ≫ 1 which are not locally finite.     

On generators and relations for unions of semigroups

Finite generation and presentability of general unions of semigroups, as well as of bands of semigroups, bands of monoids, semilattices of semigroups and strong semilattices of semigroups, are investigated. For instance, it is proved that a band Y of monoids S _α (α∈ Y ) is finitely generated/presented if and only if Y is finite and all S _α are finitely generated/presented. By way of contrast, an example is exhibited of a finitely generated semigroup which is not finitely presented, but which is a disjoint union of two finitely presented subsemigroups. January 21, 2000     

Notes on flatness and the quot functor on rings

For flat modules M over a ring A we study the similarities between the three statements,dim _k (P) ( _k (P)? _A M =dfor all prime ideals P of A, the Ap-module M _p is free of rank d for all prime ideals P of A, and M is a locally free J4-module of rank d. We have particularly emphasized the case when there is an>l-algebra B, essentially of finite type, and M is a finitely generated B-module.     

Knuth’s Coherent Presentations of Plactic Monoids of Type A

We construct finite coherent presentations of plactic monoids of type A. Such coherent presentations express a system of generators and relations for the monoid extended in a coherent way to give a family of generators of the relations amongst the relations. Such extended presentations are used for representations of monoids, in particular, it is a way to describe actions of monoids on categories. Moreover, a coherent presentation provides the first step in the computation of a categorical cofibrant replacement of a monoid. Our construction is based on a rewriting method introduced by Squier that computes a coherent presentation from a convergent one. We compute a finite coherent presentation of a plactic monoid from its column presentation and we reduce it to a Tietze equivalent one having Knuth’s generators.     

On Singular Artin Monoids and Contributions to Birman's Conjecture

ABSTRACT

Baez and Birman introduced the singular braid monoid on n + 1 strings, 𝒮B _n+1, which Birman uses in understanding knot invariants. 𝒮? _n+1 is the type A _n case of an infinite class of monoids known as singular Artin monoids and denoted by 𝒮G _M for a Coxeter matrix M. Birman conjectured, and Paris proved, that 𝒮B _n+1 embeds in the complex algebra of the braid group under the desingularisation map or Vassiliev homomorphism, η. In effect, Birman's conjecture generalizes to arbitrary types since, as noted by Corran, the Vassiliev homomorphism from 𝒮G _M to the algebra of the corresponding Artin group is well defined. We deduce general combinatorial results regarding divisibility in positive singular Artin monoids, and when M is of finite type, a well-defined positive form for 𝒮G _M is produced. These facts are then invoked to infer that, when M is of finite type, η is injective on pairs of words such that a common multiple exists for their positive form.     

Contractive presentations: A family of inverse monoids and semigroups with finiteR-classes

There has recently been considerable interest in inverse monoids which are presented by generators and relations. In this work the author employs graphical techniques to investigate the word problem for presentations of inverse monoids which generalize the case in which all relations in a presentation are of the formw=w ² . The work also investigates free objects in finitely based varieties of inverse semigroups, where the free objects have similar presentations. A fundamental charecteristic of the monoids (semigroups) investigated is: ifF is a free inverse monoid andM=F/θ, then form∈F, theR-class ofmθ has no more elements than theR-class ofm.     

Comparing Semigroup and Monoid Presentations for Finite Monoids

 It is known that for any finite group G given by a finite group presentation there exists a finite semigroup presentation for G of the same deficiency, i.e. satisfying . It is also known that the analogous statement does not hold for all finite monoids. In this paper we give a necessary and sufficient condition for a finite monoid M, given by a finite monoid presentation, to have a finite semigroup presentation of the same deficiency.     

On Finite Complete Presentations and Exact Decompositions of Semigroups

We prove that given a finite (zero) exact right decomposition (M, T) of a semigroup S, if M is defined by a finite complete presentation, then S is also defined by a finite complete presentation. Exact right decompositions are natural generalizations to semigroups of coset decompositions in groups. As a consequence, we deduce that any Zappa–Szép extension of a monoid defined by a finite complete presentation, by a finite monoid, is also defined by such a presentation.

It is also proved that a semigroup M ⁰[A; I, J; P], where A and P satisfy some very general conditions, is also defined by a finite complete presentation.     

The rees valuations of a module over a two-dimensional regular local ring

In this paper we show that the Rees valuation rings, of a finitely generated, torsion-free module M over a two-dimensional regular local ring are precisely the Rees valuation rings of the rank(M)-th Fitting invariant of M. The technical tools used are quadratic transforms and Buchsbaum-Rim multiplicity.     

Comparing Semigroup and Monoid Presentations for Finite Monoids

 It is known that for any finite group G given by a finite group presentation there exists a finite semigroup presentation for G of the same deficiency, i.e. satisfying . It is also known that the analogous statement does not hold for all finite monoids. In this paper we give a necessary and sufficient condition for a finite monoid M, given by a finite monoid presentation, to have a finite semigroup presentation of the same deficiency. (Received 17 April 2001; in revised form 15 September 2001)