10 research outputs found
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 with semilattice of projections 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 , then any right congruence on the submonoid \wt H^e_E (the \eht_E-class of ), where , can be extended to a congruence on . We introduce the idea of an {\em inverse skeleton} of a semigroup 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 , showing that \mbox{BR}(M,\theta) is a Zappa-Sz\'{e}p product of under addition and a semidirect product . We put Kunze's result in more general framework and give an analogous result for certain restriction monoids. We consider the {\it -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 -semidirect product of two restriction semigroups is restriction. introduce the notion of -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 -semidirect product of two restriction semigroups and to develop -Zappa-Sz\'{e}p products and construct a category. In the special case where is a semilattice and is a monoid we order our category to become inductive and thus obtain a restriction semigroup via the use of the standard pseudo-product
Semigroup presentations
In this thesis we consider the following two fundamental problems for semigroup presentations:
1. Given a semigroup find a presentation defining it.
2. Given a presentation describe the semigroup defined by it.
We also establish other related results.
After an introduction in Chapter 1, we consider the first problem in Chapter 2, and establish a presentation for the commutative semigroup of integers Zpt. Dually, in Chapter 3 we consider the second problem and study presentations of semigroups related to the direct product of cyclic groups. In Chapter 4 we study presentations of semigroups related to dihedral groups and establish their V-classes structure in Chapter 5. In Chapter 6 we establish some results related to the Schutzenberger group which were suggested by our studies of the semigroup presentations in Chapters 3 and 4. Finally, in Chapter 7 we define and study new classes of semigroups which we call R, L-semi-commutative and semi-commutative semigroups and they were also suggested by our studies of the semigroup presentations in Chapters 3 and 4