An orthomodular lattice $L$ is said to have fully nontrivial commutator if the commutator of any pair $x,y\in L$ is different from zero. In this note we consider the class of all orthomodular lattices with fully nontrivial commutators. We show that this class forms a quasivariety, we describe it in terms of quasiidentities and situate important types of orthomodular lattices (free lattices, Hilbertian lattices, etc.) within this class. We also show that the quasivariety in question is not a variety answering...

In the logico-algebraic foundation of quantum mechanics one often deals with the orthomodular lattices (OML) which enjoy state-separating properties of noncompatible pairs (see e.g. , and ). These properties usually guarantee reasonable “richness” of the state space—an assumption needed in developing the theory of quantum logics. In this note we consider these classes of OMLs from the universal algebra standpoint, showing, as the main result, that these classes form quasivarieties. We also illustrate...

Certain ring-like structures, so-called orthorings, are introduced which are in a natural one-to-one correspondence with lattices with 0 every principal ideal of which is an ortholattice. This correspondence generalizes the well-known bijection between Boolean rings and Boolean algebras. It turns out that orthorings have nice congruence and ideal properties.