Weakly orthomodular and dually weakly orthomodular lattices
The authors study the varieties of lattices with a unary operation~\('\) satisfying one or two or the following equations: \begin{align*} x= & (x\land y) \lor (x\land (x\land y)')\\ x= & (x\lor y) \land (x\lor (x\lor y)')\,. \end{align*} In ortholattices, any of these equations is equivalent to orthomodularity. However, here the lattice need not be bounded and the complement~\('\) need not be antitone and involutive. Nevertheless, many properties typical for orthomodular lattices can be proved also in this more general case. The corresponding varieties are arithmetical and congruence regular. Generalizations of horizontal sums, commuting elements, and decompositions to direct products are described.
scientific article; zbMATH DE number 6949481