Introduction to semigroup theory

In this paper we consider finite generation and finite presentability of Rees matrix semigroups (with or without zero) over arbitrary semigroups. The main result states that a Rees matrix semigroup M[S; I, J; P] is finitely generated (respectively, finitely presented) if and only if S is finitely generated (respectively, finitely presented), and the sets I, J and S\U are finite, where U is the ideal of S generated by the entries of P .

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Generators and relations of Rees matrix semigroups

Many classes of codes can be characterized as families of antichains with respect to partial orders on the free monoid or, in more general terms, as families of independent sets with respect to some binary relations. In this paper we investigate the general properties of this connection between families of sets and binary relations. This theory provides a framework in which known results about codes can be expressed elegantly and in which several new results are derived. Moreover, this theory can be generalized to relations of arbitrary finite arity in a very natural fashion. This allows us, for instance, to prove new hierarchy results. More importantly, however, this theory provides a new and profound insight into the mechanisms by which classes of codes are defined.

Relations on free monoids, their independent sets, and codes

The study of hairpin-free words has been initiated in the context of DNA computing. DNA strands that, theoretically speaking, are finite strings over the alphabet {A, G, C, T} are used in DNA computing to encode information. Due to the fact that A is complementary to T and G to C, DNA single strands that are complementary can bind to each other or to themselves in either intended or unintended ways. One of the structures that is usually undesirable for biocomputation, since it makes the affected DNA string unavailable for future interactions, is the hairpin: if some subsequences of a DNA single string are complementary to each other, the string will bind to itself forming a hairpin-like structure. This paper continues the theoretical study of hairpin-free languages. We study algebraic properties of hairpin-free words and hairpins. We also give a complete characterization of the syntactic monoid of the language consisting of all hairpin-free words over a given alphabet and illustrate it with an example using the DNA alphabet.

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The syntactic monoid of hairpin-free languages

We survey results and problems concerning structural properties of the hierarchy of classes of codes. In particular, we show that abstract dependence systems can be used to gain new insights into hierarchy constructions and decidability proofs. From the same perspective, we also review some work on maximal codes.

The Hierarchy of Codes

Necessary and sufficient conditions on a monoid M are found in order that M be isomorphic to the syntactic monoid of a language L over an alphabet X having one of the following properties. In the first theorem L is a PL-class and PW(L) C PL where PL is the syntactic congruence of L and W(L) is the residue of L. In the second theorem L is an infix code; that is, satisfies u, uvw e L implying u = w = 1. In the third theorem L is an infix code satisfying a condition which amounts to the requirement that M be a nilmonoid. Various refinements of these conditions are also considered.

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The syntactic monoid of an infix code

A Rees matrix semigroup over an inverse semigroup contains a greatest regular subsemigroup. The regular semigroups obtained in this manner are abstractly characterized here. The greatest completely simple homomorphic image and the idempotent generated part of such semigroups are investigated. Rectangular bands of semilattices of groups and some special cases are characterized in several ways.

Rees matrix semigroups over inverse semigroups

In this paper we investigate the semigroup structure of the syntactic monoid Syn(C+) of C+, the semigroup generated by a maximal prefix code C for which C+ is a single class of the syntactic congruence. In particular we prove that for such a prefix code C, either Syn(C+) is a group or it is isomorphic to a special type of submonoid of G× T (R) where G is a group and T (R) is the full transformation semigroup on a set R with more than one element. From this description we conclude that Syn(C+) has a kernel J which is a right group. We further investigate separately the case when J is a right zero semigroup and the case when J is a group.

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The syntactic monoid of the semigroup generated by a maximal prefix code

For a DOL-scheme (X, (,p), where K is a finite alphabet and (p is an endomorphism of X*, we study the properties of the congruence (p induced by p in terms of the properties of X*po. We prove that every submonoid of X* has a disjunctive subset (for any X) and deduce that b is a syntactic congruenlce. As special cases, we consider the conditions on (p which are equivalent to W being perfect or uniquely perfect or linear. The latter is introduced in the paper together with a ramification.

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Congruences associated with DOL-schemes

An inverse semigroup S shall be said to be harmonic if every congruence on S is determined by any one of its classes. In other words, if λ and ρ are congruences on S having a congruence class in common, then λ = ρ. The class of all harmonic semigroups contains all bisimple inverse semigroups, as proved by Žitomirskiĭ [ 11 ] and also by Schein [ 10 ], and all congruence-free inverse semigroups. Moreover, is contained in the class of all 0-simple or simple inverse semigroups, as is easy to see. We shall show that there exist non-bisimple, non-congruence-free harmonic semigroups and that there are simple inverse semigroups which are not harmonic.

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Mario Petrich

Papers

The syntactic monoid of an infix code

Rees matrix semigroups over inverse semigroups

The syntactic monoid of the semigroup generated by a maximal prefix code

Congruences associated with DOL-schemes

Harmonic inverse semigroups