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

Decomposition of semigroups into semidirect and Zappa-Sz\'{e}p products

This thesis focuses on semidirect and Zappa-Sz\'{e}p products in the context of semigroups and monoids. We present a survey of direct, semidirect and Zappa-Sz\'{e}p products and discuss correspondence between external and internal versions of these products for semigroups and monoids. Particular attention in this thesis is paid to a wide class of semigroups known as restriction semigroups. We consider Zappa-Sz\'{e}p product of a left restriction semigroup $S$ with semilattice of projections $E$ and determine algebraic properties of it. We prove that analogues of Green's lemmas and Green's theorem hold for certain semigroups where Green's relations \ar,\el,\eh and \dee are replaced by \art_E,\elt_E, \eht_E and \adt_E. We show that if \eht_E is a congruence on a certain semigroup $S$, then any right congruence on the submonoid \wt H^e_E (the \eht_E-class of $e$), where $e\in E$, can be extended to a congruence on $S$. We introduce the idea of an {\em inverse skeleton} $U$ of a semigroup $S$ and examine some conditions under which we obtain skeletons from monoids. We focus on a result of Kunze \cite{kunze:1983} for the Bruck-Reilly extension \mbox{BR}(M,\theta) of a monoid $M$, showing that \mbox{BR}(M,\theta) is a Zappa-Sz\'{e}p product of $\N^0$ under addition and a semidirect product $M\rtimes \N^0$. We put Kunze's result in more general framework and give an analogous result for certain restriction monoids. We consider the {\it $\lambda$-semidirect product} of two left restriction semigroups and prove that it is left restriction. In the two sided case using the notion of double action we prove that the $\lambda$-semidirect product of two restriction semigroups is restriction. introduce the notion of $(A,T)$-propernessto prove the results analogous to McAlister's covering theorem and O'Carroll's embedding theorem for monoids and left restriction monoids under some conditions. We extend the notion of the $\lambda$-semidirect product of two restriction semigroups $S$ and $T$ to develop $\lambda$-Zappa-Sz\'{e}p products and construct a category. In the special case where $S$ is a semilattice and $T$ is a monoid we order our category to become inductive and thus obtain a restriction semigroup via the use of the standard pseudo-product