Common Generalizations of Orthocomplete and Lattice Effect Algebras

Abstract

Connections between the weak orthocompleteness and the maximality property in effect algebras are presented. It is proved that an orthomodular poset with the maximality property is disjunctive. A characterization of Archimedean weakly orthocomplete effect algebras is given.Keywords Effect algebra · weakly orthocomplete · maximality property · disjunctive · Archimedean · separable Ovchinnikov [7] introduced weakly orthocomplete orthomodular posets (he called them alternative) as a common generalization of orthocomplete orthomodular posets and orthomodular lattices and showed that they are disjunctive. Weak orthocompleteness is useful in the study of orthoatomisticity and disjunctivity might be used to characterize atomisticity [7,11]. Weak orthocompleteness was generalized by De Simone and Navara [1].Tkadlec [8] introduced the class of orthomodular posets with the maximality property as another common generalization of orthocomplete orthomodular posets and orthomodular lattices. He showed various consequences of this property and generalize it [8,9,10,12].We show that these two notions are incomparable, that maximality property also implies disjunctivity in orthomodular posets and present a characterization of Archimedean weakly orthocomplete effect algebras. We show also some other relations within the class of effect algebras.

Homogeneous orthocomplete effect algebras are covered by MV-algebras

“…It is easy to see that an effect algebra E has the maximality property if and only if {u, v} has a maximal lower bound w, w ≥ t for every u, v, t ∈ E such that t is a lower bound of {u, v}. As noted in [19] E has the maximality property if and only if {u, v} has a minimal upper bound w for every u, v ∈ E. [19, Theorem 3.1] Let E be an Archimedean effect algebra fulfilling the condition (W+), and let y, z ∈ E. Every lower bound of y, z is below a maximal one and every upper bound of y, z is above a minimal one. Then E has the maximality property.…”

“…In [19] Tkadlec introduced the property (W+) as a common generalization of orthocomplete and lattice effect algebras.…”

Note on Generalizations of Orthocomplete and Lattice Effect Algebras

“…Weak orthocompleteness is useful in the study of orthoatomisticity and disjunctivity might be used to characterize atomisticity [4,8]. Weak orthocompleteness was generalized by De Simone and Navara [1] to the so-called property (W+), which was generalized for effect algebras by Tkadlec [10].Tkadlec [5] introduced the class of orthomodular posets with the maximality property as another common generalization of orthocomplete orthomodular posets and orthomodular lattices. He showed various consequences of this property and generalized it to the so-called property (CU) [5,6,7,9].…”

“…Weak orthocompleteness was generalized by De Simone and Navara [1] to the so-called property (W+), which was generalized for effect algebras by Tkadlec [10].…”

“…Weak orthocompleteness is useful in the study of orthoatomisticity and disjunctivity might be used to characterize atomisticity [4,8]. Weak orthocompleteness was generalized by De Simone and Navara [1] to the so-called property (W+), which was generalized for effect algebras by Tkadlec [10].…”