Abstract
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.
Highlights
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
It is well known that a lattice ordered effect algebra contains both a lattice structure and an effect algebra structure; 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 E
Summary
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. A partial ordering on an effect algebra is defined by a ≤ b if and only if there is a c ∈ E, such that a ⊕ c = b. Such an element c is unique (if it exists) and is denoted by b ⊖ a. An orthocomplementation lattice (E; ∨, ∧, , 0, 1) is called an orthomodular lattice if it satisfies the orthomodular Law: for any a, b ∈ E, a ≤ b implies b = a ∨ (a ∨ b) (1). In an orthomudular lattice, the complementation operation in a lattice is the same as the orthosupplement operation in an effect algebra
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