Decomposition of semigroups into semidirect and Zappa-Szép products
Rida-e Zenab · White Rose eTheses Online (University of Leeds, The University of Sheffield, University of York) · 2014
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.