Two Remarks to Bifullness of Centers of Archimedean Atomic Lattice Effect Algebras

M Kalina

doi:10.14311/1398

Abstract

Lattice effect algebras generalize orthomodular lattices as well as MV-algebras. This means that within lattice effect algebras it is possible to model such effects as unsharpness (fuzziness) and/or non-compatibility. The main problem is the existence of a state. There are lattice effect algebras with no state. For this reason we need some conditions that simplify checking the existence of a state. If we know that the center C(E) of an atomic Archimedean lattice effect algebra E (which is again atomic) is a bifull sublattice of E, then we are able to represent E as a subdirect product of lattice effect algebras Ei where the top element of each one of Ei is an atom of C(E). In this case it is enough if we find a state at least in one of Ei and we are able to extend this state to the whole lattice effect algebra E. In [8] an atomic lattice effect algebra E (in fact, an atomic orthomodular lattice) with atomic center C(E) was constructed, where C(E) is not a bifull sublattice of E. In this paper we show that for atomic lattice effect algebras E (atomic orthomodular lattices) neither completeness (and atomicity) of C(E) nor σ-completeness of E are sufficient conditions for C(E) to be a bifull sublattice of E.

Highlights

Lattice effect algebras generalize orthomodular lattices as well as MV-algebras
If E with respect to ≤ is lattice ordered, we say that E is a lattice effect algebra
Jenca and Riecanova in [7] proved that in every lattice effect algebra E the set S(E) = {x ∈ E; x ∧ x = 0} of sharp elements is an orthomodular lattice which is a sub-effect algebra of E, meaning that if among x, y, z ∈ E with x ⊕ y = z at least two elements are in S(E) x, y, z ∈ S(E)

Summary

Preliminaries

Definition 1 (Foulis and Bennett [3]) An effect algebra is a system (E; ⊕, 0, 1) consisting of a set E with two different elements 0 and 1, called zero and unit, respectively and ⊕ is a partially defined binary operation satisfying the following conditions for all p, q, r ∈ E: (E1) If p ⊕ q is defined, q ⊕ p is defined and p ⊕ q = q ⊕ p. Important properties of Archimedean atomic lattice effect algebras with an atomic center were proven by Riecanova in [21]. Theorem 1 (Riecanova [21]) Let E be an Archimedean atomic lattice effect algebra with an atomic center C(E).

Orthomodular lattice L whose center is not a bifull sublattice

Completion of the center of L