Orthocomplemented lattices with a symmetric difference

Abstract

Modelling an abstract version of the set-theoretic operation of symmetric difference, we first introduce the class of orthocomplemented difference lattices (ODL). We then exhibit examples of ODLs and investigate their basic properties finding, for instance, that any ODL induces an orthomodular lattice (OML) but not all OMLs can be converted to ODLs. We then analyse an appropriate version of ideals and valuations in ODLs and show that the set-representable ODLs form a variety. We finally investigate the question of constructing ODLs from Boolean algebras and obtain, as a by-product, examples of ODLs that are not set-representable but that "live" on set-representable OMLs.

Orthocomplemented difference lattices in association with generalized rings

“…The ODLs have been systematically studied in [12]- [16] where, among others, several (non-Boolean or even non-set-representable) examples of ODLs can be found. Let us list and verify the properties of ODLs that we shall use in the sequel.…”

“…The algebraic properties of ODLs have been analysed in the papers [12]- [16]. Let us first recall (see [12]) that an ODL is an algebra L = (X, ∧, ∨, ⊥ , 0, 1, ), where : X 2 → X is a binary operation and (X, ∧, ∨, ⊥ , 0, 1) is an orthocomplemented lattice.…”

“…Let us first recall (see [12]) that an ODL is an algebra L = (X, ∧, ∨, ⊥ , 0, 1, ), where : X 2 → X is a binary operation and (X, ∧, ∨, ⊥ , 0, 1) is an orthocomplemented lattice. The presence of the "symmetric difference" makes ODLs fairly close to Boolean algebras -for instance, (X, ∧, ∨, ⊥ , 0, 1) is an orthomodular lattice and (X, ) is a 2-group.…”

On identities in orthocomplemented difference lattices

“…Let us note that each Boolean algebra can be viewed as an ODL (more general ODLs will be met later, see also [10,12]). …”

“…Let us return to the ODLs that are set-representable. They form a variety ( [10]) and in view of the Stone set representation for Boolean algebras they could be seen as nearly Boolean. Though the name itself suggests their definition, let us recall it in more formal terms.…”