In this paper, a connection between rewriting systems and embedding of monoids in groups is found. We show that if a group with a positive presentation has a complete rewriting system R that satisfies the condition that each rule in R with positive left-hand side has a positive right-hand side, then the monoid presented by the subset of positive rules from R embeds in the group. As an example, ...

We describe a simple scheme for constructing finitely generated monoids in which left-divisibility is a linear ordering and for practically investigating these monoids. The approach is based on subword reversing, a general method of combinatorial group theory, and connected with Garside theory, here in a non-Noetherian context. As an application we describe several families of ordered groups wh...

Finite monoids that generate monoid varieties with uncountably many subvarieties seem rare, and surprisingly, no finite monoid is known to generate a monoid variety with countably infinitely many subvarieties. In the present article, it is shown that there are, nevertheless, many finite monoids that generate monoid varieties with continuum many subvarieties; these include any finite inherently ...

The generators of the Temperley-Lieb algebra generate a monoid with an appealing geometric representation. It has been much studied, notably by Louis Kau¤man. Borisavljevíc, Došen and Petríc gave a complete proof of its abstract presentation by generators and relations, and suggested the name ‘Kau¤man monoid’. We bring the theory of semigroups to the study of a certain …nite homomorphic image o...

We prove that the theory of root closed monoids is axiomatizable, but not finitely axiomatizable. Some directions for further research are presented. All monoids in this paper are assumed cancellative and commutative.

We present simple graph-theoretic characterizations of Cayley graphs for left-cancellative monoids, groups, left-quasigroups and quasigroups. We show that these characterizations are effective for the end-regular graphs of finite degree.

A pair (~, ~) of families of subsets of an n-element set X is cancellative if, for all A, A' e .~ and B, B' E ~, the following conditions hold: A\B = A ' \ B ~ A =A' and BkA =B'kA~B = B'. We prove that every such pair satisfies I.~11~1 < 0 ~, where 0 ~2.3264. This is related to a conjecture of ErdSs and Katona on cancellative families and to a conjecture of Simonyi on recovering pairs. For the ...

We define and study the Picard group of a monoid scheme and the class group of a normal monoid scheme. To do so, we develop some ideal theory for (pointed abelian) noetherian monoids, including primary decomposition and discrete valuations. The normalization of a monoid turns out to be a monoid scheme, but not always a monoid.

The concept of algebraic simpliication is of great importance for the eld of symbolic computation in computer algebra. In this paper we review some fundamental concepts concerning reduction rings in the spirit of Buchberger. The most important properties of reduction rings are presented. The techniques for presenting monoids or groups by string rewriting systems are used to deene several types ...