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

Acta Polytechnica

M. Kalina¹ 

关键词: lattice effect algebra;    orthomodular lattice;    center;    atom;    bifullness;   

DOI  :  

来源: Czech Technical University in Prague, Faculty of M

【 摘 要 】

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 centerC(E)of an atomic Archimedean lattice effect algebraE(which is again atomic) is a bifull sublattice ofE , then we are able to representEas a subdirect product of lattice effect algebrasE i where the top element of each one ofE i is an atom ofC(E) . In this case it is enough if we find a state at least in one ofE i and we are able to extend this state to the whole lattice effect algebraE . In [8] an atomic lattice effect algebraE(in fact, an atomic orthomodular lattice) with atomic centerC(E)was constructed, whereC(E)is not a bifull sublattice ofE . In this paper we show that for atomic lattice effect algebrasE(atomic orthomodular lattices) neither completeness (and atomicity) ofC(E)norσ -completeness ofEare sufficient conditions forC(E)to be a bifull sublattice ofE .

【 授权许可】

Unknown