Relations and Graphs

Relation Algebras We have deduced Propositions 2.3.6 and 2.3.7 from Propositions 2.3.2, 2.3.3, and 2.3.4 by arguments involving no element relationship (x,y) E R. In the exercises we require similar proofs for Dedekind's formula 2.3.5. Proofs not involving quantifiers are usually shorter and more transparent, hence also easier to check and to memorize. Moreover, this type of proof is capable of being performed by a machine. We define an (abstract) relation algebra to be an algebraic structure that satisfies the laws of an atomic, complete, Boolean lattice, the properties of multiplication (2.3.2), the Tarski rule (2.3.3), and the SchrOder equivalences (2.3.4). By contrast, when we consider a given set V and relations on it, we refer to concrete relations. We call a proof relation-algebraic, if it involves only the laws of a relation algebra, and their consequences, such as (2.3.6), but makes no use of the matrix-like structure of ReV x V. Results proved in such a way hold true in every relation algebra, hence also in those constructed by the procedure of (7.5.14).

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