The Diclique Representation and Decomposition of Binary Relations

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Abstract
The binary relation is often a useful mathematical structure for representing simple relationships whose essence is a directed connection. To better aid in interpreting or storing a binary relation we suggest a diclique decomposition. A diclique of a binary relation R is defined as an ordered pair ( I, O ) such that I × O ⊆ R and ( I, O ) is maximal. In this paper, an algorithm is described for determining the dicliques of a binary relation; it is proved that the set of such dicliques has a nice algebraic structure. The algebraic structure is used to show how dicliques can be coalesced, the relationship between cliques and dicliques is discussed, and an algorithm for determining cliques from dicliques is described.

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