On presentations of Brauer-type monoids

We obtain presentations for the Brauer monoid, the partial analogue of the Brauer monoid, and for the greatest factorizable inverse submonoid of the dual symmetric inverse monoid. In all three cases we apply the same approach, based on the realization of all these monoids as Brauer-type monoids.     

Catalan monoids,monoids of local endomorphisms,and their presentations

The Catalan monoid and partial Catalan monoid of a directed graph are introduced. Also introduced is the notion of a local endomorphism of a tree, and it is shown that the Catalan (resp. partial Catalan) monoid of a tree is simply its monoid of extensive local endomorphisms (resp. partial endomorphisms) of finite shift. The main results of this paper are presentations for the Catalan and partial Catalan monoids of a tree. Our presentation for the Catalan monoid of a tree is used to give an alternative proof for a result of Higgins. We also identify results of Aîzen?tat and Popova which give presentations for the Catalan monoid and partial Catalan monoid of a finite symmetric chain.     

Minimal presentations for monoids with the ascending chain condition on principal ideals

We show that the natural way to extend several key results concerning minimal presentations for finitely generated commutative cancellative reduced monoids, is to replace the finitely generated condition by the ascending chain condition on principal ideals.     

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.     

Minimal but inefficient presentations of the semi-direct products of some monoids

In recent papers of Ruskuc, Saito and J. Wang, the semi-direct product of two arbitrary monoids and a standard presentation, say P, for this product have received considerable attention. Wang defined a trivialiser set of the Squier complex associated with P and after that necessary and sufficient conditions for P to be efficient have been given by Cevik. As a main result of this paper, we give sufficient conditions for a presentation of the semi-direct product of a one-relator monoid by an infinite cyclic monoid to be minimal but not efficient. In the final part of this paper we give some applications of this result.     

Morita duality for monoids

In this paper Morita duality for monoids is introduced. Necessary and sufficient conditions for two monoids S and T to be Morita dual are given. Moreover, it is shown that if S and T are Morita dual monoids, then S and U are Morita dual if and only if T and U are Morita equivalent. In addition, every finite monoid having Morita duality is selfdual and even reflexive.     

Endomorphism monoids of acts over monoids

In this paper we investigate under which conditions a monoid R is defined by the endomorphism monoid of an act over R. More precisely, we ask when an isomorphism between two such endomorphism monoids over monoids R₁ and R₂ is induced by a semilinear isomorphism. The question is considered also for ordered and for topological monoids. On the way we characterize monoids over which all projective acts are free. An abstract of this paper appeared in the Proceedings of the Conference on Semigroups, Szeged 1972.     

Right-orderability versus left-orderability for monoids

Semigroup Forum - We investigate the relationship between (total) left- and right-orderability for monoids, in particular illustrating the finite case by various structural observations and...     

Endomorphism monoids of free acts and O-wreath products of monoids

Monoids and acts which may have zero elements are considered. In Section 1 we construct a O-wreath product of monoids. In 2 we prove the theorem that the endomorphism monoid of a free act over a monoid with zero can be represented as a O-wreath product. Considering monoids with tero we are interested in their annihilator properties. In 3 we give necessary and sufficient conditions for a O-wreath product of monoids to be a right (left) Baer (Rickart) monoid. In 4 we obtain as a consequence corresponding conditions for the endomorphism monoid of a free act over a monoid with zero.     

Graph monoids

A graph semigroup refers to a monoid whose defining relations are of the form x_ix_j=x_jx_i. We describe the centralizer of an arbitrary element of a graph semigroup, show that there exists a unique factorization of any element into commuting parts, and prove related results. Dedicated to L. M. Gluskin on the occasion of his sixtieth birthday     

Hereditary and semi-hereditary monoids

Left hereditary and semi-hereditary monoids are analogous to left hereditary and semi-hereditary rings. Necessary and sufficient conditions are given for the heredity and semi-heredity of arbitrary and commutative monoids.     

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.