Internalizing equality in Boolean algebras (Q1866807)

An equality algebra is a universal algebra \(A\) together with a semilattice \((L,\cdot ,1)\) and a binary function \(=_i : A\times A \rightarrow L\) satisfying the reflexivity rule \((x=_i x)=1\) and the replacement rule \((x=_i y)f(x)=(x=_i y)f(y)\) for all functions \(f:A\rightarrow L\) derived from the operations on \(A\), the semilattice operation and \(=_i\). The most important case the authors consider is that one in which the Boolean ring \(R\) has \(=_i\) taking values in the semilattice \((R,\cdot ,1)\). This is called an EB-ring. It is shown that the variety of EB-rings is equivalent to the variety of modal rings (Boolean rings with an interior operation). Varying the strength and exact nature of the replacement property corresponds to selecting from a number of natural varieties of modal rings.