On algebras of relations with operations of left and right reflexive product (Q2207015)

This paper investigates the classes of ordered and unordered algebras, \((A, \ast, \subseteq)\) or \((A, \ast)\), where \(A\) is a set of binary relations on some set, \(\subseteq\) is set-theoretical inclusion, and the binary operation \(\ast\) is defined by \[ \rho\ast \sigma = \{(u,v)\;:\;(\exists w)((u,u)\in\rho\;\wedge (w,w)\in \sigma)\}. \] The class of ordered algebras of this type is denoted \(R\{\ast,\subseteq\}\) and the class of unordered algebras of this type is denoted \(R\{\ast\}\). The paper provides axiomatizations for the classes \(R\{\ast,\subseteq\}\) and \(R\{\ast\}\). It is shown that neither of these classes is a quasivariety, but the quasivariety generated by either class is a variety, and axioms are provided for these varieties.