A Note of Filters in Effect Algebras

We investigate relations of the two classes of filters in effect algebras (resp., MV-algebras). We prove that a lattice filter in a lattice ordered effect algebra (resp., MV-algebra) does not need to be an effect algebra filter (resp., MV-filter). In general, in MV-algebras, every MV-filter is also a lattice filter. Every lattice filter in a lattice ordered effect algebra is an effect algebra filter if and only if is an orthomodular lattice. Every lattice filter in an MV-algebra is an MV-filter if and only if is a Boolean algebra. 1. Introduction The notion of effect algebras has been introduced by Foulis and Bennett [1] as an algebraic structure providing an instrument for studying quantum effects that may be unsharp. One of D-posets has been introduced by Chovanec and K？pka [2]. The two notions are categorically equivalent. Many results with respect to effect algebras and D-posets have been obtained (see [1–6]). A comprehensive introduction about effect algebras can been found in the monograph [7]. The filter theory of effect algebras is an important objects of investigation (see [3, 4, 8, 9]). It is well known that a lattice ordered effect algebra contains both a lattice structure and an effect algebra structure; hence, the notions of lattice filters and effect algebra filters are investigated, respectively. One would ask: what relations are there between lattice filters and effect algebra filters? In this paper we discuss this problem in a lattice ordered effect algebra (resp., an MV-algebra). A lattice filter in a lattice ordered effect algebra does not need to be an effect algebra filter (resp., MV-filter). In general, lattice filter in a lattice ordered effect algebra is an effect algebra filter if and only if is an orthomodular lattice. A lattice filter in an MV-algebra is an MV-filter if and only if is a Boolean algebra. Definition 1 (see [1]). An effect algebra is an algebraic structure , where is a nonempty set, and are distinct elements of , and is a partial binary operation on that satisfies the following conditions.(E1) Commutative Law. If is defined, then is defined and .(E2) Associative Law. If and are defined, then and are defined and .(E3) Orthosupplementation Law. For any , there is a unique , such that .(E4) Zero-One Law. If is defined, then . When the hypotheses of (E2) are satisfied, we write for element in . For simplicity, we use the notation for an effect algebra. If is defined, we write and whenever we write we are implicitly assuming that . A partial ordering on an effect algebra is defined by if and only if there is a , such