Some properties of congruence relations on orthomodular lattices. (English) Zbl 1011.06011

Dorfer, Gerhard

Discuss. Math., Gen. Algebra Appl. 21, No. 1, 57-66 (2001).

Briefly speaking, an orthomodular lattice (OML) is a Boolean algebra where the distributive law is substituted by the weaker orthomodular law \(x\vee y=x\vee((x\vee y)\wedge x')\). The paper under review deals with OMLs. The structure of the lattice of \(p\)-ideals (which are in a bijective correspondence to congruence relations) as well as the structure of congruence classes is investigated. From these results the well-known fact that OMLs are congruence regular, congruence uniform and arithmetical is derived.

Reviewer: Helmut Länger (Wien)

MSC:

06C15

Complemented lattices, orthocomplemented lattices and posets

Keywords:

orthomodular lattice; congruence relation; \(p\)-ideal; congruence class; congruence regular; congruence uniform; arithmetical