scholarly journals A Note of Filters in Effect Algebras

2013 ◽  
Vol 2013 ◽  
pp. 1-4
Author(s):  
Biao Long Meng ◽  
Xiao Long Xin
Keyword(s):  
Boolean Algebra ◽  
Orthomodular Lattice ◽  
Effect Algebra ◽  
Effect Algebras ◽  
Lattice Filter ◽  
Mv Algebra ◽  
Mv Algebras

We investigate relations of the two classes of filters in effect algebras (resp., MV-algebras). We prove that a lattice filter in a lattice ordered effect algebra (resp., MV-algebra) does not need to be an effect algebra filter (resp., MV-filter). In general, in MV-algebras, every MV-filter is also a lattice filter. Every lattice filter in a lattice ordered effect algebra is an effect algebra filter if and only if is an orthomodular lattice. Every lattice filter in an MV-algebra is an MV-filter if and only if is a Boolean algebra.

The pasting constructions for effect algebras

2014 ◽  
Vol 64 (5) ◽  
Author(s):  
Yongjan Xie ◽  
Yongming Li ◽  
Aili Yang
Keyword(s):  
Boolean Algebra ◽  
Orthomodular Lattice ◽  
Effect Algebra ◽  
Finite Lattice ◽  
Effect Algebras ◽  
The Family

AbstractThe aim of this paper is to present several techniques of constructing a lattice-ordered effect algebra from a given family of lattice-ordered effect algebras, and to study the structure of finite lattice-ordered effect algebras. Firstly, we prove that any finite MV-effect algebra can be obtained by substituting the atoms of some Boolean algebra by linear MV-effect algebras. Then some conditions which can guarantee that the pasting of a family of effect algebras is an effect algebra are provided. At last, we prove that any finite lattice-ordered effect algebra E without atoms of type 2 can be obtained by substituting the atoms of some orthomodular lattice by linear MV-effect algebras. Furthermore, we give a way how to paste a lattice-ordered effect algebra from the family of MV-effect algebras.

Effect-like algebras induced by means of basic algebras

2010 ◽  
Vol 60 (1) ◽  
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.

scholarly journals Study of MV-algebras via derivations

2019 ◽  
Vol 27 (3) ◽  
pp. 259-278
Author(s):  
Jun Tao Wang ◽  
Yan Hong She ◽  
Ting Qian
Keyword(s):  
Boolean Algebra ◽  
Direct Product ◽  
Galois Connection ◽  
Product Representation ◽  
Mv Algebra ◽  
One To One ◽  
The Relationship ◽  
Mv Algebras ◽  
Derivation Pair

AbstractThe main goal of this paper is to give some representations of MV-algebras in terms of derivations. In this paper, we investigate some properties of implicative and difference derivations and give their characterizations in MV-algebras. Then, we show that every Boolean algebra (idempotent MV-algebra) is isomorphic to the algebra of all implicative derivations and obtain that a direct product representation of MV-algebra by implicative derivations. Moreover, we prove that regular implicative and difference derivations on MV-algebras are in one to one correspondence and show that the relationship between the regular derivation pair (d, g) and the Galois connection, where d and g are regular difference and implicative derivation on L, respectively. Finally, we obtain that regular difference derivations coincide with direct product decompositions of MV-algebras.

scholarly journals Effect algebras with state operator

10.29007/jbdq ◽  
2018 ◽  
Author(s):  
Silvia Pulmannova
Keyword(s):  
Effect Algebra ◽  
Operator Algebras ◽  
Internal State ◽  
Effect Algebras ◽  
State Operator ◽  
Conditional Expectations ◽  
Definition Of ◽  
Mv Algebras

A state operator on effect algebras is introduced as an additive, idempotent and unital mapping from the effect algebra into itself. The definition is inspired by the definition of an internal state on MV-algebras, recently introduced by Flaminio and Montagna. We study state operators on convex effect algebras, and show their relations with conditional expectations on operator algebras.

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

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.

On realization of effect algebras

2016 ◽  
Vol 66 (2) ◽  
Author(s):  
Josef Niederle ◽  
Jan Paseka
Keyword(s):  
Quantum Logic ◽  
Orthomodular Lattice ◽  
Effect Algebra ◽  
Simplex Algorithm ◽  
Effect Algebras ◽  
Standard Quantum ◽  
Generalized Effect Algebra ◽  
Determining Set

AbstractA well known fact is that there is a finite orthomodular lattice with an order determining set of states which is not order embeddable into the standard quantum logic, the latticeWe show that a finite generalized effect algebra is order embeddable into the standard effect algebraAs an application we obtain an algorithm, which is based on the simplex algorithm, deciding whether such an order embedding exists and, if the answer is positive, constructing it.

scholarly journals The block structure of complete lattice ordered effect algebras

2007 ◽  
Vol 83 (2) ◽  
pp. 181-216 ◽  
Author(s):  
Gejza Jenča
Keyword(s):  
Complete Lattice ◽  
Orthomodular Lattice ◽  
Effect Algebra ◽  
Block Structure ◽  
Effect Algebras

AbstractWe prove that every for every complete lattice-ordered effect algebra E there exists an orthomodular lattice O(E) and a surjective full morphism øE: O(E) → E which preserves blocks in both directions: the (pre)imageofa block is always a block. Moreover, there is a 0, 1-lattice embedding : E → O(E).

A triple representation of lattice effect algebras

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.

scholarly journals Central elements and Cantor-Bernstein's theorem for pseudo-effect algebras

2003 ◽  
Vol 74 (1) ◽  
pp. 121-144 ◽  
Author(s):  
Anatolij Dvurečenskij
Keyword(s):  
Effect Algebras ◽  
Riesz Decomposition Property ◽  
Bernstein Theorem ◽  
Riesz Decomposition ◽  
Mv Algebra ◽  
Central Elements ◽  
Partial Algebras ◽  
Left And Right ◽  
The Riesz Decomposition Property ◽  
Mv Algebras

AbstractPseudo-effect algebras are partial algebras (E; +, 0, 1) with a partially defined addition + which is not necessary commutative and with two complements, left and right ones. We define central elements of a pseudo-effect algebra and the centre, which in the case of MV-algebras coincides with the set of Boolean elements and in the case of effect algebras with the Riesz decomposition property central elements are only characteristic elements. If E satisfies general comparability, then E is a pseudo MV-algebra. Finally, we apply central elements to obtain a variation of the Cantor-Bernstein theorem for pseudo-effect algebras.

scholarly journals Effect algebras, witness pairs and observables

10.29007/71gb ◽  
2018 ◽  
Author(s):  
Gejza Jenča
Keyword(s):  
Boolean Algebra ◽  
Open Problem ◽  
Effect Algebra ◽  
Effect Algebras ◽  
Orthomodular Posets ◽  
Interval Effect ◽  
Bounded Posets

The category of effect algebras is the Eilenberg-Moore category for themonad arising from the free-forgetful adjunction between categories of bounded posetsand orthomodular posets.In the category of effect algebras, an observable is a morphism whose domainis a Boolean algebra. The characterization of subsets of ranges of observables isan open problem.For an interval effect algebra E, a witness pair for a subset of S is an objectliving within E that "witnesses existence" of an observable whose rangeincludes S. We prove that there is an adjunction between the poset of allwitness pairs of E and the category of all partially inverted E-valuedobservables.

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