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}.
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机译：本文主要讨论集合X上二元关系的半群B {dollar} sb {lcub} rm X {rcub} {dollar}中的主要理想之间的关系。我们假设X始终是有限的。大多数工作是通过将二进制关系表示为布尔矩阵来完成的。在本文的开头，我们给出了矩阵表示传递关系和等价关系的条件。然后，当{dollar} vert {dollar} X {dollar} vert {dollar} = 0，1，2时，我们描述B {dollar} sb {lcub} rm X {rcub} {dollar}中的一组主要理想的结构。 ，3和4;并描述当{dollar} vert {dollar} X {dollar} vert {dollar} = 0、1、2和3时B {dollar} sb {lcub} rm X {rcub} {dollar}中所有理想的集合。给出了所有主要理想集合中链的最大长度的估计。由于前面提到的集合的大小很大，因此本文稍后将进行一些尝试来简化B {dollar} sb {lcub} rm X {rcub} {dollar}中主要理想之间关系的确定。我们给一个主理想提供了必要的条件，以使其能够覆盖一组主理想中的另一个主理想，并使用这些条件来找到涵盖并由某些特殊主理想覆盖的主理想的数量。 } rm X {rcub} {dollar}。
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