Dynamic effect algebras

AbstractWe introduce the so-called tense operators in lattice effect algebras. Tense operators express the quantifiers “it is always going to be the case that” and “it has always been the case that” and hence enable us to express the dimension of time in the logic of quantum mechanics. We present an axiomatization of these tense operators and prove that every lattice effect algebra whose underlying lattice is complete can be equipped with tense operators. Such an effect algebra is called dynamic since it reflects changes of quantum events from past to future.

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STATES, UNIFORMITIES AND METRICS ON LATTICE EFFECT ALGEBRAS

International Journal of Uncertainty Fuzziness and Knowledge-Based Systems ◽

10.1142/s0218488502001879 ◽

2002 ◽

Vol 10 (supp01) ◽

pp. 125-133 ◽

Cited By ~ 11

Author(s):

ZDENKA RIEČANOVÁ

Keyword(s):

Effect Algebra ◽

Order Topology ◽

Effect Algebras ◽

Lattice Effect Algebra ◽

Uniform Topology ◽

Lattice Effect

We show that every state ω on a lattice effect algebra E induces a uniform topology on E. If ω is subadditive this topology coincides with pseudometric topology induced by ω. Further, we show relations between the interval and order topology on E and topologies induced by states.

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Two Remarks to Bifullness of Centers of Archimedean Atomic Lattice Effect Algebras

Acta Polytechnica ◽

10.14311/1398 ◽

2011 ◽

Vol 51 (4) ◽

Author(s):

M. Kalina

Keyword(s):

Effect Algebra ◽

Subdirect Product ◽

Sufficient Conditions ◽

Effect Algebras ◽

Lattice Effect Algebra ◽

Orthomodular Lattices ◽

Atomic Lattice ◽

Lattice Effect ◽

Archimedean Lattice ◽

Mv Algebras

Lattice effect algebras generalize orthomodular lattices as well as MV-algebras. This means that within lattice effect algebras it is possible to model such effects as unsharpness (fuzziness) and/or non-compatibility. The main problem is the existence of a state. There are lattice effect algebras with no state. For this reason we need some conditions that simplify checking the existence of a state. If we know that the center C(E) of an atomic Archimedean lattice effect algebra E (which is again atomic) is a bifull sublattice of E, then we are able to represent E as a subdirect product of lattice effect algebras Ei where the top element of each one of Ei is an atom of C(E). In this case it is enough if we find a state at least in one of Ei and we are able to extend this state to the whole lattice effect algebra E. In [8] an atomic lattice effect algebra E (in fact, an atomic orthomodular lattice) with atomic center C(E) was constructed, where C(E) is not a bifull sublattice of E. In this paper we show that for atomic lattice effect algebras E (atomic orthomodular lattices) neither completeness (and atomicity) of C(E) nor σ-completeness of E are sufficient conditions for C(E) to be a bifull sublattice of E.

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Sharply Orthocomplete Effect Algebras

Acta Polytechnica ◽

10.14311/1267 ◽

2010 ◽

Vol 50 (5) ◽

Author(s):

M. Kalina ◽

J. Paseka ◽

Z. Riečanová

Keyword(s):

Effect Algebra ◽

Effect Algebras ◽

Lattice Effect Algebra ◽

Atomic Lattice ◽

Complete Lattices ◽

Lattice Effect

Special types of effect algebras E called sharply dominating and S-dominating were introduced by S. Gudder in [7, 8]. We prove statements about connections between sharp orthocompleteness, sharp dominancy and completeness of E. Namely we prove that in every sharply orthocomplete S-dominating effect algebra E the set of sharp elements and the center of E are complete lattices bifull in E. If an Archimedean atomic lattice effect algebra E is sharply orthocomplete then it is complete.

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A triple representation of lattice effect algebras

Mathematica Slovaca ◽

10.1515/ms-2016-0221 ◽

2016 ◽

Vol 66 (6) ◽

Author(s):

Ivan Chajda ◽

Helmut Länger

Keyword(s):

Effect Algebra ◽

Effect Algebras ◽

Mutual Relationship ◽

Lattice Effect Algebra ◽

Antitone Involution ◽

Lattice Effect ◽

One To One ◽

Mv Algebras

AbstractA mutual relationship between MV-algebras and coupled semirings as established by L. P. Belluce, A. Di Nola, A. R. Ferraioli and B. Gerla is extended to lattice effect algebras and so-called characterizing triples. We show that this correspondence is in fact one-to-one and hence every lattice effect algebra can be considered as an ordered triple consisting of two semiring-like structures and an antitone involution which is an isomorphism between these structures.

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The sum of observables on a $$\sigma $$ σ -distributive lattice effect algebra

Soft Computing ◽

10.1007/s00500-018-3617-8 ◽

2018 ◽

Vol 23 (16) ◽

pp. 6743-6753

Author(s):

Jiří Janda ◽

Yongming Li

Keyword(s):

Distributive Lattice ◽

Effect Algebra ◽

Lattice Effect Algebra ◽

Lattice Effect ◽

Sum Of Observables

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Effect-like algebras induced by means of basic algebras

Mathematica Slovaca ◽

10.2478/s12175-009-0164-x ◽

2010 ◽

Vol 60 (1) ◽

Cited By ~ 3

Author(s):

Ivan Chajda

Keyword(s):

Distributive Lattice ◽

Effect Algebra ◽

Binary Operation ◽

Basic Algebra ◽

Natural Question ◽

Lattice Effect Algebra ◽

Mv Algebra ◽

Lattice Effect ◽

Mv Algebras

AbstractHaving an MV-algebra, we can restrict its binary operation addition only to the pairs of orthogonal elements. The resulting structure is known as an effect algebra, precisely distributive lattice effect algebra. Basic algebras were introduced as a generalization of MV-algebras. Hence, there is a natural question what an effect-like algebra can be reached by the above mentioned construction if an MV-algebra is replaced by a basic algebra. This is answered in the paper and properties of these effect-like algebras are studied.

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Properties of implication in effect algebras

Mathematica Slovaca ◽

10.1515/ms-2021-0001 ◽

2021 ◽

Vol 71 (3) ◽

pp. 523-534

Author(s):

Ivan Chajda ◽

Helmut Länger

Keyword(s):

Hilbert Space ◽

Quantum Mechanics ◽

State Space ◽

Effect Algebra ◽

Modus Ponens ◽

Effect Algebras ◽

Algebraic Semantics ◽

Deductive Systems ◽

Algebraic Description ◽

Residuated Poset

Abstract Effect algebras form a formal algebraic description of the structure of the so-called effects in a Hilbert space which serve as an event-state space for effects in quantum mechanics. This is why effect algebras are considered as logics of quantum mechanics, more precisely as an algebraic semantics of these logics. Because every productive logic is equipped with implication, we introduce here such a concept and demonstrate its properties. In particular, we show that this implication is connected with conjunction via a certain “unsharp” residuation which is formulated on the basis of a strict unsharp residuated poset. Though this structure is rather complicated, it can be converted back into an effect algebra and hence it is sound. Further, we study the Modus Ponens rule for this implication by means of so-called deductive systems and finally we study the contraposition law.

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