The semigroup of binary relations

This paper mainly deals with the relationship between principal ideals in B{dollar}sb{lcub}rm X{rcub}{dollar}, the semigroup of binary relations on a set X. We assume X to be finite throughout. Most of the work was done by representing a binary relation as a Boolean matrix.; Early in the dissertation we give conditions under which a matrix represents a transitive relation and an equivalence relation. We then describe the structure of the set of principal ideals in B{dollar}sb{lcub}rm X{rcub}{dollar} when {dollar}vert{dollar}X{dollar}vert{dollar} = 0, 1, 2, 3 and 4; and describe the set of all ideals in B{dollar}sb{lcub}rm X{rcub}{dollar} when {dollar}vert{dollar}X{dollar}vert{dollar} = 0, 1, 2 and 3. An estimate of the maximal length of a chain in the set of all principal ideals is given.; Because of the large size of the previously mentioned sets, some attempt at simplifying the determination of the relationship between principal ideals in B{dollar}sb{lcub}rm X{rcub}{dollar} is given later in the paper. We give necessary conditions for a principal ideal to cover another principal ideal in the set of principal ideals, and use these conditions to find the number of principal ideals which cover, and are covered by some special principal ideals of B{dollar}sb{lcub}rm X{rcub}{dollar}.