Elsevier

Information Sciences

Volume 179, Issue 5, 15 February 2009, Pages 529-534
Information Sciences

Pseudocomplemented lattice effect algebras and existence of states

https://doi.org/10.1016/j.ins.2008.07.019Get rights and content

Abstract

We prove that in every pseudocomplemented atomic lattice effect algebra the subset of all pseudocomplements is a Boolean algebra including the set of sharp elements as a subalgebra. As an application, we show families of effect algebras for which the existence of a pseudocomplementation implies the existence of states. These states can be obtained by smearing of states existing on the Boolean algebra of sharp elements.

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 x 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 Sh(E) 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 S(E) of pseudocomplements is a Boolean algebra with at least one atom and including Sh(E) 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 (P;,0) with zero 0 and we will write aPb=0 if 0 is a unique common lower bound of {a,b}P.

Definition 2.1

Let (P;,0) be a poset with zero 0. An element aP is called a pseudocomplement of aP if:

  • (i)

    aPa=0,

  • (ii)

    aPx=0xa.

If every element of P has a pseudocomplement then P is called a pseudocomplemented poset with zero.

We can easily check that each element aP 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 (E;,0,1) 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 a,b,cE:

  • (Ei)

    ba=ab if ab is defined,

  • (Eii)

    (ab)c=a(bc) if one side is defined,

  • (Eiii)

    for every aE there exists a unique bE such that ab=1 (we put a=b),

  • (Eiv)

    if 1a is defined then a=0.

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.

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