Abstract
Special types of effect algebras E called sharply dominating and S-dominating were introduced by S. Gudder in [7, 8]. We prove statements about connections between sharp orthocompleteness, sharp dominancy and completeness of E. Namely we prove that in every sharply orthocomplete S-dominating effect algebra E the set of sharp elements and the center of E are complete lattices bifull in E. If an Archimedean atomic lattice effect algebra E is sharply orthocomplete then it is complete.
Highlights
An algebraic structure called an effect algebra was introduced by D
A lattice effect algebra E is an orthomodular lattice iff every element of E is sharp (i.e., x and “non x” are disjoint) and it is an MV-effect algebra iff every pair of elements of E is compatible
In every lattice effect algebra E the set of sharp elements is an orthomodular lattice ([10]), and E is a union of its blocks (i.e., maximal subsets of pairwise compatible elements that are MV-effect algebras)
Summary
An algebraic structure called an effect algebra was introduced by D. In every lattice effect algebra E the set of sharp elements is an orthomodular lattice ([10]), and E is a union of its blocks (i.e., maximal subsets of pairwise compatible elements that are MV-effect algebras (see [21])). S-dominating effect algebras may be useful abstract models for sets of quantum effects in physical systems. We study these two special kinds of effect algebras. We show properties of some remarkable subeffect algebras of such effect algebras E satisfying the condition that E is sharply orthocomplete Properties of their blocks, sets of sharp elements and their centers. It is worth noting that it was proved in [11] that there are even Archimedean atomic MVeffect algebras which are not sharply dominating, they are not S-dominating
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