Lattice effect algebras possessing two-valued states
by
Jan Paseka
Department of Mathematics and Statistics, Faculty of Science, Masaryk University, Janackovo nam. 2a, 602 00 Brno, Czech Republic
Coauthors: Zdena Riecanova, Department of Mathematics, Faculty of Electrical Engineering and Information Technology, Slovak University of Technology Ilkovicova 3, SK-812 19Bratislava, Slovak Republic, zdena.riecanova@gmail.com

Common generalizations of MV-algebras [] and orthomodular lattices are lattice effect algebras []. An effect algebra (E;⊕, 0, 1) is a set E with two special elements 0, 1 and a partial binary operation ⊕ which is commutative and associatiove at which these equalities hold if one of their sides exists. Moreover, to every element a ∈ E there exists a unique element a' ∈ E with a⊕a'=1 and if a⊕1 exists then a=0. In every effect algebra we can define a partial order by a ≤ b iff there exists c ∈ E with a⊕c=b (we set c=b\ominus a). If (E; ≤ ) is a lattice (a complete lattice) then (E;⊕, 0, 1) is called a lattice effect algebra (a complete lattice effect algebra).

Generalized effect algebras as posets are unbounded versions of effect algebras. In this case instead of the axiom on the existence of a' with a⊕a'=1 for all a ∈ E we have cancellation law, i.e., a⊕b=a⊕c implies b=c and, moreover, a⊕b=0 implies a=b=0.

A well known fact is that every generalized effect algebra P can be uniquely extended onto effect algebra E (called an effect algebraic extension of P) in which P is an order ideal in E and P^*=E\P is a dual poset to P. We write E=P[(∪)\dot] P^* (a disjoint union) []. On the other hand not every (lattice) effect algebra E becomes this way. We can prove

[] Let (E;⊕, 0, 1) be an effect algebra. The following conditions are equivalent:

(i) There exists a two-valued state w on E.
(ii) There exists a sub-generalized effect algebra P_w of E such that E=P_w[(∪)\dot] P_w^*, where P_w^*={1\ominus x | x ∈ P_w}, P_w∩P_w^*=∅ and P_w is an order ideal in E.

For Archimedean atomic lattice effect algebra E we can show a sufficient condition (F) for the existence of a two-valued state w on E:

(F) There exists a finite set F={p_k | k ∈ H} of pairwise noncompatible atoms of E such that for every atomic block M of E there exists k_M ∈ H such that p_kM ∈ C(M) and w(p_kM)=1. []

Moreover, we can prove that this condition (F) is a necessary and sufficient condition for the existence of a two-valued state w on: (a) every (o)-continuous Archimedean atomic lattice effect algebra, (b) every block-finite Archimedean atomic lattice effect algebra. Here a block of a lattice effect algebra is a maximal sub-lattice effect algebra being an MV-algebra. A lattice effect algebra is called block-finite if it has only finitely many blocks.

Using this facts we can prove

[] Every Archimedean atomic lattice effect algebra with at most five blocks possess a state.

Note that in present time the known example of finite lattice effect algebra admitting no states has nineteen blocks.

Finally, since the existence of a two-valued state w on an effect algebra E is equivalent to the fact that E is an effect algebraic extension of a sub-generalized effect algebra P_w={x ∈ E | w(x)=0}, the following question arises:

If F₁ and F₂ are two sets of pairwise noncompatible atoms of an Archimedean atomic lattice effect algebra E and w₁, w₂ are two-valued states on E at which P_F1=w^-1₁({0}) and P_F2=w^-1₂({0}) are sub-generalized effect algebras of E with E=P_F1[(∪)\dot] P^*_F1=P_F2[(∪)\dot] P^*_F2 whether (or at which conditions) P_F1 and P_F2 are isomorphic generalized effect algebras.

We show that two non-isomorphic generalized effect algebras P_F1 and P_F2 may have a common (or isomorphic) effect algebraic extension.

In spite of the fact that an atomic lattice effect algebra E may have nonatomic block [] the following statement can be proved:

[] Let E be an Archimedean atomic lattice effect algebra and F₁, F₂ are two sets of pairwise noncompatible atoms satisfying the condition (F). Let there exists a bijection y:F₁→ F₂ such that, for every atomic block M of E we have a ∈ M∩F₁ ⇔ y(a) ∈ M∩F₂. Then P_F1 ≅ P_F2.

This very simple condition is not necessary for the isomorphism of P_F1 and P_F2. The necessary and sufficient condition for P_F1 and P_F2 is the following:

[] Let E₁, E₂ be Archimedean atomic lattice effect algebras, F₁ ⊆ E₁, F₂ ⊆ E₂ are two sets of pairwise noncompatible atoms satisfying the condition (F). Let M₁, M₂ are families of all atomic blocks of E₁ and E₂ respectively. Then the condition (i) implies the conditions (ii) and (iii):

(i)

P_F1 and P_F2 are isomorphic generalized effect algebras.
(ii) There exist bijections y:At(E₁)→ At(E₂) and a: M₁→ M₂ such that for any M ∈ M₁ there exists an isomorphism j_M:M→ a(M) of atomic MV-effect algebras such that j_M(a)=y(a) for all a ∈ At(M) and y(F₁)=F₂.
(iii) There is a bijection y:At(E₁)→ At(E₂) such that



(a)

y(F₁)=F₂.
(b) ord(y(a))=ord(a) for all a ∈ At(E₁).
(c) a↔ b iff y(a)↔ y(b), for all atoms a, b ∈ At(E₁).

References

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E.G. Beltrametti, G. Cassinelli, The Logic of Quantum Mechanics, Addison-Wesley, Reading, MA, 1981.

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C.C. Chang, Algebraic analysis of many-valued logics, Trans. Amer. Math. Soc. 88 (1958) 467-490.

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D.J. Foulis, M.K. Bennett, Effect algebras and unsharp quantum logics, Found. Phys. 24 (1994), 1325-1346.

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J. Hedlíková, S. Pulmannová, Generalized difference posets and orthoalgebras, Acta Math. Univ. Comenianae 45 (1996) 247-279.

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J. Paseka, Z. Riecanová, On atoms based isomorphism theorems of lattice or generalized prelattice effect algebras, preprint, 2007.

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Z. Riecanová: Effect algebraic extensions of generalized effect algebras and two-valued states, Fuzzy Sets and Systems, to appear.

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Z. Riecanová, The existence of states on every Archimedean atomic lattice effect algebra with at most five blocks, Kybernetika, to appear.

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Date received: February 22, 2008