Brunetto Piochi

Amalgams of free inverse semigroups

We study inverse semigroup amalgams of the formS *U T whereS andT are free inverse semigroups andU is an arbitrary finitely generated inverse subsemigroup ofS andT. We make use of recent work of Bennett to show that the word problem is decidable for any such amalgam. This is in contrast to the general situation for semigroup amalgams, where recent work of Birget, Margolis and Meakin shows that the word problem for a semigroup amalgamS *U T is in general undecidable, even ifS andT have decidable word problem,U is a free semigroup, and the membership problem forU inS andT is decidable. We also obtain a number of results concerning the structure of such amalgams. We obtain conditions for theD-classes of such an amalgam to be finite and we show that the amalgam is combinatorial in such a case. For example every one-relator amalgam of this type has finiteD-classes and is combinatorial. We also obtain information concerning when such an amalgam isE-unitary: for example every one relator amalgam of the formInv whereA andB are disjoint andu (resp.v) is a cyclically reduced word overA ∪A−1 (resp.B ∪B−1) isE-unitary.

A combinatorial approach to collapsing words

Given a word w over a finite alphabet E and a finite deterministic automaton A = (Q,Σ,δ), the inequality |δ(Q,ω}| < |Q| - n means that under the natural action of the word w the image of the state set Q is reduced by at least n states. The word w is n-collapsing if this inequality holds for any deterministic finite automaton that satisfies such an inequality for at least one word. In this paper we present a new approach to the topic of collapsing words, and announce a few results we have obtained using this new approach. In particular, we present a direct proof of the fact that the language of n-collapsing words is recursive.

Logical matrices and non-structural consequence operators

In the present paper, we study some properties of matrices for non-structural consequence operators. These matrices were introduced in a former work (see [3]). In sections 1. and 2., general definitions and theorems are recalled; in section 3. a correspondence is studied, among our matrices and Wójcickis ones for structural operators. In section 4. a theorem is given about operators, induced by submatrices or epimorphic images, or quotient matrices of a given one.Such matrices are used to characterize lattices of non-structural consequence operators, by constructing lattices, antiisomorphic to them (see section 5.). In the last section, a sufficient condition is given for a non-structural operator to be finite.

Rank and Status in Semigroup Theory

Abstract For each generating set A of a finite semigroup S the integer Δ(A) is defined as the least n for which every element of S is expressible as a product of at most n elements of A. The status of S is defined as the least value of |A|Δ(A) among generating sets of A. Some general bounds are obtained, and the notion is explored in more detail for certain well understood classes of semigroups.

Congruences on Bruck-Reilly extensions of monoids

We characterize the three general types of congruences on a Bruck-Reilly monoid; hence an explicit description is also given for the homomorphic closure of the class of Bruck-Reilly extensions.

Permutability of normal extensions of a semilattice

LetS be a semigroup;S is said to bepermutable if, for some integern, every product ofn elements ofS can be re-ordered. We prove that every normal extension of a semilattice by an inverse permutable semigroupsis permutable. Also, some properties of permutable groups are extended to inverse semigroups.

On fixed-point-free permutation properties in groups and semigroups

AbstractWe say that a semigroup S is (fixed-point-free, for short f.p.f.) permutable, if, for some integer n and for every x1,..., xn in S, there exists a non-trivial (fixed-point-free) permutation σ on {1,..., n}, such that:

A finiteness condition for semigroups which generalizes Green-Rees' theorem

x_1 \ldots x_n = x_{\sigma (1)} \ldots x_{\sigma (n)} .