An atomic MV-effect algebra with non-atomic center

A set E equipped with a partial, commutative and associative operation ⊕, containing elements 0 and 1, in which the existence of a unique inverse element x′ to any x ∈ E is guaranteed, and a⊕ 1 is admitted only if a = 0 is well known as effect algebra. Effect algebras were introduced by Foulis and Bennet [3] and simultaneously by Kôpka and Chovanec [7] as D-posets. An effect algebra equipped with partial order 6 can form a lattice called a lattice effect algebra. This structure generalizes both orthomodular lattices, i. e. the effect algebras in which x⊕ x is not defined for any non-zero x ∈ E, and MV-effect algebras, i. e. effect algebras with all pairs of elements being compatible [6], and it is applied as a carrier of probability of unsharp or fuzzy events. In connection with existence of states on lattice effect algebras properties of the subalgebra of sharp elements and the subalgebra of central elements are studied. The following definitions are consistent with the ones in [2].