Pseudocomplemented lattice effect algebras and existence of states☆
Introduction
Common generalizations of orthomodular lattices which may include noncompatible pairs of elements and MV-algebras including unsharp elements are lattice effect algebras. They were introduced for modeling sets of propositions, properties, questions, or events with fuzziness, uncertainty or unsharpness. Effect algebras are very natural logical structures as carriers of probabilities and states. Nevertheless, there are even finite effect algebras admitting no states and probabilities [8]. The existence of states on effect algebras is an open question. Only some families of effect algebras which posses states are known (modular or interval effect algebras, see [10], [1]). We are going to show some further families of lattice effect algebras, the existence of a pseudocomplementation on which implies the existence of states.
Elements of lattices or posetes with zero may have only uniquely determined pseudocomplements. Nevertheless, a pseudocomplementation on a poset P need not be an orthocomplementation on P, namely for quantum structures with unsharp elements (meaning that elements x and (non)x, hence x and need not be disjoint). We show that for all pseudocomplemented lattice effect algebras their sets of sharp elements are sub-lattices, on which inherited pseudocomplementation and orthocomplementation coincide. In consequence, for every pseudocomplemented lattice effect algebra E, the set of its sharp elements is a Boolean algebra which is a sublattice effect algebra of E. Moreover, if E is atomic non-MV-effect algebra then the set of pseudocomplements is a Boolean algebra with at least one atom and including as a subalgebra. As an application, we can show some families of lattice effect algebras such that the existence of a pseudocomplementation implies the existence of states on them (Section 4).
Section snippets
Pseudocomplemented posets with zero
In this section we will consider a poset with zero 0 and we will write if 0 is a unique common lower bound of . Definition 2.1 Let be a poset with zero 0. An element is called a pseudocomplement of if: , .
If every element of P has a pseudocomplement then P is called a pseudocomplemented poset with zero.
We can easily check that each element may have at most one pseudocomplement. Some results for pseudocomplemented posets with 0 can be found in [2], [4]. Example 2.2
Pseudocomplemented lattice effect algebras
The definition of an effect algebra is due to Foulis and Bennett [3]. Definition 3.1 A partial algebra is called an effect algebra if 0, 1 are two distinct elements and is a partially defined binary operation on E which satisfies the following conditions for any : if is defined, if one side is defined, for every there exists a unique such that (we put ), if is defined then .
In fact, every effect algebra E is a bounded poset if we define the
Existence of states on pseudocomplemented atomic lattice effect algebras
Important examples of atomic complete lattice effect algebras which are pseudocomplemented are complete atomic MV-effect algebras and distributive effect algebras. All of them possess states. A natural question arrises whether states exist on all pseudocomplemented complete atomic effect algebras. We are going to show that the answer is positive. Another family of atomic lattice effect algebras on which the existence of pseudocomplementation implies the existence of states are block-finite
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This research was supported by the Slovak Research and Development Agency under the Contract No. APVV-0071-06 and Grant VEGA-1/3025/06 of MŠSR.