Relativizations of relation algebras by the diversity
The relative subalgebra \({\mathbf A}_{0a}\) of a Boolean algebra \({\mathbf A}_0 = (A,+, \cdot, ^-,0,1)\) by an element \(a\in A\) is defined by \({\mathbf A}_{0a} = (Aa,+, \cdot, ^{-a}, 0, a)\), where \(Aa= \{x\in A:x \leq a\}\) and \(x^{-a} = x^- \cdot a\) for \(x\in A\). The relative subalgebra \({\mathbf A} a\) of an augmented Boolean algebra \({\mathbf A} = ({\mathbf A}_0, F)\) is defined by \({\mathbf A} a= ({\mathbf A}_{0a},Fa)\), where \(Fa= \{f_a:f \in F\}\) and for each \(f\in F\), \(f_a\) is defined by \(f_a(x_1, \dots, x_n) =f(x_1, \dots, x_n) \cdot a\) for \(x_1, \dots, x_n\in Aa\). Now let \({\mathbf A} = ({\mathbf A}_0, ;,^\smile, 1')\) be a relation algebra; the element \(0'= 1'{}^-\) is called the diversity element of \({\mathbf A}\). The author proves that the relative subalgebras \({\mathbf A} 0'\) form a finitely based variety. It is also proved that every relation algebra \({\mathbf A}\) is determined by \({\mathbf A} 0'\), up to a direct factor that is Boolean (the relative product coincides with the meet). An open problem is pointed out.
scientific article; zbMATH DE number 908010