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

#### 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*E*where the top element of each one of_{i}*E*is an atom of_{i}*C(E)*. In this case it is enough if we find a state at least in one of*E*and we are able to extend this state to the whole lattice effect algebra_{i}*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*.#### Keywords

lattice effect algebra; orthomodular lattice; center; atom; bifullness

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