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 ...

In this paper we combine results from [1] and [3] in order to determine the relationship between the deficiency of a group G and the deficiency of a Rees matrix semigroup S over G; in particular, we investigate how the efficiency of S depends on the efficiency of G. We start by defining semigroup and group presentations. Let A be an alphabet. We denote by A the free semigroup on A consisting of...

We give necessary and sufficient conditions for a certain class of local arithmetical congruence monoids (or ACMs) to have accepted elasticity. Let a and b be positive integers with a ≤ b and a ≡ a (mod b). Under these conditions, the arithmetic sequence a, a+ b, a+ 2b, a+ 3b, . . . is closed under multiplication and is a multiplicative semigroup. Setting M(a, b) = {a+ kb | k ∈ N0} ∪ {1} produc...

For a given reachable automaton A, we prove that the (state-)endomorphism monoid End(A) divides its characteristic monoid M(A). Hence so does its (state-)automorphism group Aut(A), and, for finite A, Aut(A) is a homomorphic image of a subgroup of the characteristic monoid. It follows that in the presence of a (state-) automorphism group G of A, a finite automaton A (and its transformation monoi...

We describe Green’s relations on the partition monoid, two of its submonoids, and the related Jones and Brauer monoids. Key Words: partition monoid, Jones monoid, ideal structure 2000 Mathematics Subject Classi…cation: 20M20 1 The partition monoid and some submonoids The elements of the partition monoid Pn [5, 8, 9] are equivalence classes of graphs on a vertex set n [ n0 = f1; 2; :::ng [ f10; ...