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...

We construct an extension E(A,G) of a given group G by infinite nonArchimedean words over an discretely ordered abelian group like Z. This yields an effective and uniform method to study various groups that ”behave like G”. We show that the Word Problem for f.g. subgroups in the extension is decidable if and only if and only if the Cyclic Membership Problem in G is decidable. The present paper ...

We introduce a new algebraic construction, monop, that combines monoids (with respect to the product of species), and operads (monoids with respect to the substitution of species) in the same algebraic structure. By the use of properties of cancellative set-monops we construct a family of partially ordered sets whose prototypical examples are the Dowling lattices. They generalize the partition ...

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 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...

We give an algorithmic method for computing a presentation of any finitely generated submonoid of a finitely generated commutative monoid. We use this method also for calculating the intersection of two congruences on Np and for deciding whether or not a given finitely generated commutative monoid is t-torsion free and/or separative. The last section is devoted to the resolution of some simple ...

where both terms consist of two alternating letters and have the same length. First investigated by J.Tits in the late 1960s [2], and then in [3] and [11], these groups remain incompletely understood, with many open questions, including the decidability of the Word Problem in the general case [6]. The only well understood case is the one of spherical type, which is the case when the associated ...