On a non-trivial partially ordered real vector space (V,≤) the orthogonality relation is defined by incomparability and ζ(V,⊥) is a complete lattice of double orthoclosed sets. We say that A ⊆ V is an orthogonal set when for all a,b ∈ A with a ≠ b, we have a ⊥ b. In our earlier papers we defined an integrally open ordered vector space and two closure operations A → D(A) and $A\to A^{\perp \perp }$. It was proved that V is integrally open iff $D\left(A\right)=A^{\perp \perp }$ for every orthogonal set A ⊆ V. In this paper we generalize this result. We...

The paper deals with orthomodular lattices which are so-called horizontal sums of Boolean algebras. It is elementary that every such orthomodular lattice is simple and its blocks are just these Boolean algebras. Hence, the commutativity relation plays a key role and enables us to classify these orthomodular lattices. Moreover, this relation is closely related to the binary commutator which is a term function. Using the class $\mathscr{H}$ of horizontal sums of Boolean algebras, we establish an identity which...

The set $A$ of all atoms of an atomic orthomodular lattice is said to be almost orthogonal if the set $\{b\in A:b\nleqq a^{\text{'}}\}$ is finite for every $a\in A$. It is said to be strongly almost orthogonal if, for every $a\in A$, any sequence $b_1,b_2,\cdots$ of atoms such that $a\nleqq b_1^{\text{'}},b_1\nleqq b_2^{\text{'}},\cdots$ contains at most finitely many distinct elements. We study the relation and consequences of these notions. We show among others that a complete atomic orthomodular lattice is a compact topological one if and only if the set of all its atoms is almost orthogonal.

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...

It is proved that orthomodular posets are in a natural one-to-one correspondence with certain residuated structures.

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.