Implication in weakly and dually weakly orthomodular lattices

Abstract

Weakly orthomodular and dually weakly orthomodular lattices were introduced by the authors in a recent paper. Similarly as for orthomodular lattices we try to introduce an implication in these lattices which can be easily axiomatized and which yields a nice lattice structure. As shown in the paper, this can be realized in several different ways. Moreover, we reveal the connection of weakly and dually weakly orthomodular lattices to residuated structures. Moreover, we provide a characterization of these lattices by means of certain generalized measures.