Around the relative center property in orthomodular lattices

Abstract

The paper deals with the relative center property in orthomodular lattices (OMLs). The property holds in a large class of OMLs, including locally modular OMLs and projection lattices of AW ∗ {{\text {AW}}^*} - and W ∗ {{\text {W}}^*} -algebras, and it means that the center of any interval [ 0 , a ] [0,a] is the set { a ∧ c | c central in L } \{ a \wedge c|c\;{\text {central}}\;{\text {in}}\;L\} . In §l we study the congruence lattice of an OML satisfying the Axiom of Comparability (A.C.) and, in §2, we prove that the central cover of an element can be expressed in many different ways in OMLs satisfying a certain condition (C). For complete OMLs, Axiom (A.C.) and condition (C) are equivalent to the relative center property. In §3, we give a coordinatization theorem for complete OMLs with the relative center property.