Congruence kernels of orthomodular implication algebras (Q942101)

Orthomodular implication algebras are algebras \({\mathcal A} = (A, \cdot, 1)\) satisfying axioms (expressed by identities) obtained by abstracting from certain properties of the classical implication in Boolean algebras. The authors prove that the variety of orthomodular implication algebras is weakly regular (i.e. \([1]\Theta = [1]\Phi\) implies \(\Theta = \Phi\) for all congruences \(\Theta, \Phi\)), permutable (i.e.\ \([1](\Theta \circ \Phi) = [1](\Phi \circ \Theta)\) for all congruences \(\Theta, \Phi\)), and 3-permutable (i.e.\ \(\Theta \circ \Phi \circ \Theta = \Phi \circ \Theta \circ \Phi\) for all congruences \(\Theta, \Phi\)). An orthomodular lattice is an algebra \((L, \vee, \wedge, ', 0, 1)\), where \((L, \vee, \wedge, 0, 1)\) is a bounded lattice and the unary operation \('\) satisfies the following axioms: \[ \begin{aligned} x \vee x' = 1 & (x \vee y)' = x' \wedge y' \quad x \leq y \text{ implies } y = x \vee (y \wedge x')\\ x \wedge x' = 0 & (x \wedge y)' = x' \vee y' \end{aligned} \] Orthomodular join-semilattices are partial algebras \((A, \vee, (^x; x \in A), 1)\) where \((A, \vee, 1)\) is a join-semilattice and for every \(x \in A\), \(^x\) is a unary operation on \([x, 1]\) such that \(([x, 1], \vee, \wedge_x, ^x, x, 1)\) is an orthomodular lattice. Compatible congruence families on orthomodular join-semilattices are defined, and one-to-one correspondences are established between: {\parindent=6mm \begin{itemize}\item[(1)] orthomodular implication algebras \({\mathcal A}\) and orthomodular join-semilattices \({\mathcal S}({\mathcal A})\); \item[(2)] congruences on \({\mathcal A}\) and compatible congruence families on \({\mathcal S}({\mathcal A})\). \end{itemize}} Next, the notion of a congruence kernel of an orthomodular implication algebra is introduced (a subset \(F \subseteq A\) with the property that there exists a congruence \(\Theta\) with \([1]\Theta = F\)). If \(L\) is an orthomodular lattice, two elements \(a ,b \in L\) are called perspective to each other if they have a common complement. A subset of \(L\) is called \(p\)-filter if it is closed w.r.t.\ perspectivity. One-to-one correspondences are established between: {\parindent=6mm \begin{itemize}\item[(1)] congruences on orthomodular implication algebras and their kernels; \item[(2)] congruences on and \(p\)-filters of orthomodular lattices; \item[(3)] congruence kernels of orthomodular implication algebras and compatible filter families on the corresponding orthomodular join-semilattices. \end{itemize}} The last correspondence above allows the authors to give a characterization of congruence kernels of orthomodular implication algebras.