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AAA76 - 76th Workshop on General Algebra (76. Arbeitstagung Allgemeine Algebra)
May 22-25, 2008
Department of Algebra, Johannes Kepler University Linz
Linz, Austria

Organizers
Erhard Aichinger, Peter Mayr, Matt Nickodemus, Günter Pilz

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Compactly Generated Archimedean Atomic Lattice Effect Algebras
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

Abstract
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).

Definition 1(1) An element p of an effect algebra E is called an atom if 0 ≠ b ≤ p ⇒ b=p. E is called atomic if for every x ∈ E\{0} there is an atom p ∈ E such that p ≤ x.

(2) An element u ∈ E is called e-finite if there is a finite sequence {p1, p2, ..., pn} of not necessarily different atoms of E such that u=p1⊕p2⊕...⊕pn.

(3) An e-finite element u ∈ E is called s-finite if u=⊕G=∨{⊕K | K ⊆ G is finite} implies G is finite; here G=(ak)k ∈ H is a ⊕-orthogonal system of not necessarily different atoms.

(4) An element v ∈ E is called e-cofinite (s-cofinite ) if v' is e-finite (s-finite).

Note that any atom of E is evidently s-finite and the smallest element of E is s-finite.

Definition 2 (1) An element a of a lattice L is called compact iff, for any D ⊆ L, a ≤ ∨D implies a ≤ ∨F for some finite F ⊆ D.

(2) A lattice L is called compactly generated iff every element of L is a join of compact elements.

Theorem 1 Let E be an atomic Archimedean lattice effect algebra. Then the following conditions are equivalent:


    (i) E is (o)-continuous.
    (ii) Every atom of E is compact.
    (iii) Every element of E is s-finite if and only if it is compact.
    (iv) Every element of E is e-finite if and only if it is compact.
    (v) E is a compactly generated lattice (by e-finite elements).

Theorem 2 Let E be an (o)-continuous atomic Archimedean lattice effect algebra. Then


    (i) E1={x ∈ E | x is e-finite or e-cofinite} is a sub-lattice effect algebra of E.
    (ii) E1 is sharply dominating.

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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A. Dvurecenskij, S. Pulmannová: New Trends in Quantum Structures, Kluwer Acad. Publ., Dordrecht/Ister Science, Bratislava 2000.

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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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S. P. Gudder, Sharply dominating effect algebras, Tatra Mt. Math. Publ. 15 (1998), 23-30

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Z. Riecanová, Wu Junde, States on sharply dominating effect algebras, Science in China Series A:Mathematics 51, No.5, 907-914.

Date received: May 12, 2008

Copyright © 2008 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cawc-57.