A given monoid usually admits many presentations by generators and relations the notion of Tietze equivalence characterizes when two describe same monoid: it is case one can transform presentation into other using families so-called transformations. The goal this article to provide an abstract geometrical understanding well-known fact, constructing a model structure on category presentations, i...

In this paper we are interested in the Chinese monoid and show that the Chinese monoid has complete rewriting system. Mathematics Subject Classification: 16S15; 20F05; 20F10; 20M50; 68Q42

The goal of the article is to define the concept of monoid. In the preliminary section we introduce the notion of some properties of binary operations. The second section is concerning with structures with a set and a binary operation on this set: there is introduced the notion corresponding to the notion of some properties of binary operations and there are shown some useful clusters. Next, we...

The set of homology cobordisms from a surface to itself with markings of their boundaries has a natural monoid structure. To investigate the structure of this monoid, we define and study its Magnus representation and Reidemeister torsion invariants by generalizing Kirk-Livingston-Wang’s argument over the Gassner representation of string links. Moreover, by applying Cochran and Harvey’s framewor...

We construct a recursive formula for a complete system of primitive orthogonal idempotents for any Rtrivial monoid. This uses the newly proved equivalence between the notions of R-trivial monoid and weakly ordered monoid. Résumé. Nous construisons une formule récursive pour un système complet d’idempotents orthogonaux primitifs pour tout monoı̈de R-trivial. Nous employons une nouvelle équivalenc...

We consider the problem of random uniform generation of traces (the elements of a free partially commutative monoid) in light of the uniform measure on the boundary at infinity of the associated monoid. We obtain a product decomposition of the uniform measure at infinity if the trace monoid has several irreducible components—a case where other notions such as Parry measures, are not defined. Ra...

A nite semigroup (respectively monoid or group) S is said to be eecient if it can be deened by a semigroup (respectively monoid or group) presentation hA jRi with jRj ? jAj = rank(H 2 (S 1)). In this paper we show that a group is eecient as a group if and only if it is eecient as a monoid, but some eecient monoids are not eecient as semigroups. Moreover we show that certain classes of eecient g...