\(MP\)-algebras with relative types (Q1272216)

An \(MP\)-algebra with relative types \((B,s,t_1,t_2,t_3)\) is a Boolean algebra \(B\) with four unitary maps \(s\), \(t_1\), \(t_2\), \(t_3\) from \(B\) to itself s.t. \((B,s)\) is a monadic algebra [see \textit{P. R. Halmos,} Algebraic logic (1962; Zbl 0101.01101)] satisfying the following conditions: \(t_i \big (p\cap s(q)\big)= t_i(p)\cap s(q),\) for \(i=1,2,3\), \(s(t_i(p))=t_i(p),\) for \(i=1,2,3\), \(s(p)\subset t_1(p)\cup t_2(p)\cup t_3(p)\) and if \(i\not=j\), \(t_i(p)\cap t_j(p)=0\), if \(p\subset q\) then \(t_3(p)\subset t_3(q)\) and \(t_2(p)\cup t_3(p)\subset t_2(q)\cup t_3(q)\), if \(p\cap q=0\) then \(t_i(p)\cap t_j(q)\subset t_k(p\cup q)\), where \(k=\min(i+j,3)\). Every relation algebra has an \(MP\)-algebra with relative types associated with it in a natural way. The authors prove by Givant's result [see \textit{S. R. Givant}, The structure of relation algebras generated by relativizations (1994; Zbl 0812.03035)] that every \(MP\)-algebra with relative types arises in this way from some relation algebra generated by its rectangles.