On finite complete presentations and exact decompositions of semigroups

We prove that given a finite (zero) exact right decomposition (M, T) of a semigroup S, if M is defined by a finite complete presentation, then S is also defined by a finite complete presentation. Exact right decompositions are natural generalizations to semigroups of coset decompositions in groups. As a consequence, we deduce that any Zappa–Szép extension of a monoid defined by a finite complete presentation, by a finite monoid, is also defined by such a presentation. It is also proved that a semigroup M 0[A; I, J; P], where A and P satisfy some very general conditions, is also defined by a finite complete presentation

Direct products and elementary equivalence of polycyclic-by-finite groups

We give an algebraic characterization of elementary equivalence for polycyclic-by-finite groups. Using this characterization, we investigate the relations between their elementary equivalence and the elementary equivalence of the factors in their decompositions in direct products of indecomposable groups. In particular we prove that the elementary equivalence of two such groups G,H is equivalent to each of the following properties: 1)Gx...xG (k times G) and Hx...xG (k times H) are elementarily equivalent for a strictly positive integer k; 2)AxG and AxH are elementarily equivalent for two elementarily equivalent polycyclic-by-finite groups A,B. It is not presently known if 1) implies elementary equivalence for any groups G,H.Comment: 15 pages. Minor changes in pages 1 to 3, following the remarks of a referee. The paper is presently publishe

Lie semigroups with triple decompositions

In this paper we establish that the causal order determined by an Ol\u27shanski semigroup on the corresponding homogeneous space is globally hyperbolic. Using this fact, we present sufficient conditions for a special class of Lie semigroups to admit a canonical triple decomposition, namely those for which the Lie algebra is of Cayley type. This theory applies in particular to semigroups which are naturally associated to euclidean Jordan algebras as the semigroup of compressions of the symmetric cone of the algebra

Crossed products by twisted partial actions and graded algebras

For a twisted partial action \Theta of a group G on an (associative non-necessarily unital) algebra A over a commutative unital ring k, the crossed product A X_\Theta G is proved to be associative. Given a G-graded k-algebra B = \oplus_{g\in G}\B_g with the mild restriction of homogeneous non-degeneracy, a criteria is established for B to be isomorphic to the crossed product B_1 X_\Theta G for some twisted partial action of G on B_1. The equality B_g\B_{g^{-1}}B_g = \B_g for all g\in G is one of the ingredients of the criteria, and if it holds and, moreover, B has enough local units, then it is shown that B is stably isomorphic to a crossed product by a twisted partial action of G.Comment: 38 pages, no figure