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Around the relative center property in orthomodular lattices. (English) Zbl 0795.06010

Summary: This 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 \(AW^*\)- and \(W^*\)- algebras, and it means that the center of any interval \([0,a]\) is the set \(\{a\land c\): \(c\) central in \(L\}\). In §1 we study the congruence lattice of an OML satisfying the axiom of comparability (AC) 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, AC and C are equivalent to the relative center property. In §3, we give a coordinatization theorem for complete OMLs with the relative center property.

MSC:

06C15 Complemented lattices, orthocomplemented lattices and posets
20M99 Semigroups
46K99 Topological (rings and) algebras with an involution
46L99 Selfadjoint operator algebras (\(C^*\)-algebras, von Neumann (\(W^*\)-) algebras, etc.)
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