We show that the monoid M2(T) of 2 × 2 tropical matrices is a regular semigroup satisfying the semigroup identity A2B4A2A2B2A2B4A2 =A2B4A2B2A2A2B4A2. Studying reduced identities for subsemigroups of M2(T), and introducing a faithful semigroup representation for the bicyclic monoid by 2 × 2 tropical matrices, we reprove Adjan’s identity for the bicyclic monoid in a much simpler way.

Informally, a homotopy monoid is a monoid-like structure in which properties such as associativity only hold ‘up to homotopy’ in some consistent way. This short paper comprises a rigorous definition of homotopy monoid and a brief analysis of some examples. It is a much-abbreviated version of the paper ‘Homotopy Algebras for Operads’ (math.QA/0002180), and does not assume any knowledge of operads.

We prove that any affine algebraic monoid can be obtained as the endomorphisms’ monoid of a finite-dimensional (nonassociative) algebra.

Let X∗ be a free monoid over an alphabet X and W be a finite language over X. Let S(W ) be the Rees quotient X∗/I(W ), where I(W ) is the ideal of X∗ consisting of all elements of X∗ that are not subwords of W . Then S(W ) is a finite monoid with zero and is called the discrete syntactic monoid of W . W is called finitely based if the monoid S(W ) is finitely based. In this paper, we give some ...

The aim of this paper is to investigate whether the class of automaton semigroups is closed under certain semigroup constructions. We prove that the free product of two automaton semigroups that contain left identities is again an automaton semigroup. We also show that the class of automaton semigroups is closed under the combined operation of ‘free product followed by adjoining an identity’. W...

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 non-parametric, or re-sampling, bootstrap for a single unstructured sample corresponds to the algebraic operation of monoid composition, with a uniform distribution on the monoid. With this interpretation, the notion of re-sampling can be extended to designs having a certain group-invariance property. Two types of exchangeable array structures are considered in some detail, namely the one-w...

We prove several complexity and decidability results for automatic monoids: (i) there exists an automatic monoid with a P-complete word problem, (ii) there exists an automatic monoid such that the firstorder theory of the corresponding Cayley-graph is not elementary decidable, and (iii) there exists an automatic monoid such that reachability in the corresponding Cayley-graph is undecidable. Mor...

A (not necessarily commutative) Krull monoid—as introduced by Wauters—is defined as a completely integrally closed monoid satisfying the ascending chain condition on divisorial two-sided ideals. We study the structure of these Krull monoids, both with ideal theoretic and with divisor theoretic methods. Among others we characterize normalizing Krull monoids by divisor theories. Based on these re...

We say that a rational (resp. a subsequential) function tl from a free monoid into another one is in the variety of monoids V if it may realized by some unambiguous (resp. subsequential) transducer whose monoid of transitions is in V. We characterize these functions when V is the variety of aperiodic monoids, and the variety of groups. In the first case, the period of am ’ (15) divides that of ...