This thesis presents a partition of the class of homomorphisms between groupoids of n-tuples in a system g = (G,&,@), where G = { a,b,c,d,e }is a set of five elements such that: 1) a is the &-identity and annihilates all elements under @; 2) b is the @-identity; 3) d absorbs all all elements except e under & and all elements except a and e under @; 4) e absorbs all elements under & and all elements except a under @; 5) & is a binary operation on G and is commutative in G; 6) @ is a binary operation on G and is left-distributive over & in G.Matrices over g were examined for characteristics which would determine different atomic properties of homomorphisms. A matrix operation @ was defined, which allowed the homomorphisms of groupoids of the form, (G(n) , &), to be modeled by a matrix equation. Using the atomic proper ties, a partition of the class of homomorphisms between groupoids was developed, and an example of an element in each of its disjoint subsets was presented. A listing of theorems was also derived.Ball State UniversityMuncie, IN 47306

This project deals with the automorphism group G of a Steiner system S (3, 4, 10). S₁₀, the symmetrical group of degree 10, acts transitively on T, the set of all Steiner systems with parameters 3, 4, 10. The purpose of this project is to study the action of S₁₀ on cosets of G. This will be achieved by means of a graph of S₁₀ on T x T. The orbits of S₁₀ on T x T are in one-one correspondence with the orbits of G, the stabilizer of an S [e] T on T.

In this thesis we study two different topics, both in the context of semigroup constructions. The first is the investigation of an embedding problem, specifically the problem of whether it is possible to embed any given finitely presentable semigroup into a D-simple finitely presentable semigroup. We consider some well-known semigroup constructions, investigating their properties to determine whether they might prove useful for finding a solution to our problem. We carry out a more detailed study into a more complicated semigroup construction, the Byleen extension, which has been used to solve several other embedding problems. We prove several results regarding the structure of this extension, finding necessary and sufficient conditions for an extension to be D-simple and a very strong necessary condition for an extension to be finitely presentable. The second topic covered in this thesis is relative rank, specifically the sequence obtained by taking the rank of incremental direct powers of a given semigroup modulo the diagonal subsemigroup. We investigate the relative rank sequences of infinite Cartesian products of groups and of semigroups. We characterise all semigroups for which the relative rank sequence of an infinite Cartesian product is finite, and show that if the sequence is finite then it is bounded above by a logarithmic function. We will find sufficient conditions for the relative rank sequence of an infinite Cartesian product to be logarithmic, and sufficient conditions for it to be constant. Chapter 4 ends with the introduction of a new topic, relative presentability, which follows naturally from the topic of relative rank.