Relation algebras as residuated Boolean algebras
After some general remarks on residuated and conjugate operations in a Boolean algebra, a residuated Boolean algebra is defined as a Boolean algebra that is augmented with three binary operations, \(\circ\), \(\vartriangleleft\), and \(\vartriangleright\), such that for all elements \(x\), \(y\), and \(z\), \((x\circ y)\cdot z= 0\) iff \((x\vartriangleright z)\cdot 0\) iff \((z\vartriangleleft y)\cdot x= 0\). Such an algebra is unital if there is an identity element \(e\) for \(\circ\). Every nonassociative relation algebra gives rise to a unital residuated Boolean algebra by defining \(x\circ y= x;y\), \(x\vartriangleleft y= x;\breve y\) and \(x\vartriangleleft y= \breve x;y\). Several characterizations are given for those unital residuated Boolean algebras arising in this way. Several equations equivalent to the associativity of \(\circ\) are given, along with characterizations of those unital residuated Boolean algebras that come from relation algebras.
scientific article; zbMATH DE number 488263