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 Quantum B-algebras with involutions

2020 ◽  
pp. 2150233
Author(s):  
Lavinia Corina Ciungu
Keyword(s):  
Algebraic Structures ◽  
Wajsberg Hoops ◽  
Residuated Poset ◽  
One To One ◽  
Universal Quantifiers ◽  
The Relationship ◽  
Mv Algebras

The aim of this paper is to define and study the involutive and weakly involutive quantum B-algebras. We prove that any weakly involutive quantum B-algebra is a residuated poset. As an application, we introduce and investigate the notions of existential and universal quantifiers on involutive quantum B-algebras. It is proved that there is a one-to-one correspondence between the quantifiers on weakly involutive quantum B-algebras. One of the main results consists of proving that any pair of quantifiers is a monadic operator on weakly involutive quantum B-algebras. We investigate the relationship between quantifiers on bounded sup-commutative pseudo BCK-algebras and quantifiers on other related algebraic structures, such as pseudo MV-algebras and bounded Wajsberg hoops.

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.

Quantum computational algebra with a non-commutative generalization

2016 ◽  
Vol 66 (1) ◽  
Author(s):  
Wenjuan Chen ◽  
Wieslaw A. Dudek
Keyword(s):  
Direct Product ◽  
Computational Algebra ◽  
Product Decomposition ◽  
Direct Product Decomposition ◽  
Mv Algebra ◽  
Mv Algebras

AbstractWe introduce a non-commutative generalization of quasi-MV algebra, called quasipseudo-MV algebra. We present some properties of quasi-pseudo-MV algebras and investigate the direct product decomposition of them. Further, we generalize quasi

An example of a commutative basic algebra which is not an MV-algebra

2010 ◽  
Vol 60 (2) ◽  
Author(s):  
Michal Botur
Keyword(s):  
Subdirect Product ◽  
Lattice Structure ◽  
Basic Algebra ◽  
Commutative Basic Algebra ◽  
Mv Algebra ◽  
One To One ◽  
Open Question ◽  
Mv Algebras

AbstractMany algebras arising in logic have a lattice structure with intervals being equipped with antitone involutions. It has been proved in [CHK1] that these lattices are in a one-to-one correspondence with so-called basic algebras. In the recent papers [BOTUR, M.—HALAŠ, R.: Finite commutative basic algebras are MV-algebras, J. Mult.-Valued Logic Soft Comput. (To appear)]. and [BOTUR, M.—HALAŠ, R.: Complete commutative basic algebras, Order 24 (2007), 89–105] we have proved that every finite commutative basic algebra is an MV-algebra, and that every complete commutative basic algebra is a subdirect product of chains. The paper solves in negative the open question posed in [BOTUR, M.—HALAŠ, R.: Complete commutative basic algebras, Order 24 (2007), 89–105] whether every commutative basic algebra on the interval [0, 1] of the reals has to be an MV-algebra.

scholarly journals Ideals in residuated lattices

2021 ◽  
Vol 37 (1) ◽  
pp. 53-63
Author(s):  
DUMITRU BUŞNEG ◽  
DANA PICIU ◽  
ANCA-MARIA DINA
Keyword(s):  
Fuzzy Sets ◽  
Binary Relation ◽  
Congruence Relation ◽  
Natural Generalization ◽  
Residuated Lattices ◽  
Algebraic Analysis ◽  
Deductive Systems ◽  
Mv Algebra ◽  
The Relationship ◽  
Mv Algebras

"The notion of ideal in residuated lattices is introduced in [Kengne, P. C., Koguep, B. B., Akume, D. and Lele, C., L-fuzzy ideals of residuated lattices, Discuss. Math. Gen. Algebra Appl., 39 (2019), No. 2, 181–201] and [Liu, Y., Qin, Y., Qin, X. and Xu, Y., Ideals and fuzzy ideals in residuated lattices, Int. J. Math. Learn & Cyber., 8 (2017), 239–253] as a natural generalization of that of ideal in MV algebras (see [Cignoli, R., D’Ottaviano, I. M. L. and Mundici, D., Algebraic Foundations of many-valued Reasoning, Trends in Logic-Studia Logica Library 7, Dordrecht: Kluwer Academic Publishers, 2000] and [Chang, C. C., Algebraic analysis of many-valued logic, Trans. Amer. Math. Soc., 88 (1958), 467–490]). If A is an MV algebra and I is an ideal on A then the binary relation x ∼I y iff x^{*}Ꙩ y; x Ꙩy^{*} ∈ I , for x; y ∈ A; is a congruence relation on A. In this paper we find classes of residuated lattices for which the relation ∼ I (defined for MV algebras) is a congruence relation and we give new characterizations for i-ideals and prime i-ideals in residuated lattices. As a generalization of the case of BL algebras (see [Lele, C. and Nganou, J. B., MV-algebras derived from ideals in BL-algebras, Fuzzy Sets and Systems, 218 (2013), 103–113]), we investigate the relationship between i-ideals and deductive systems in residuated lattices."

K-Radical classes and product radical classes of MV-algebras

2008 ◽  
Vol 58 (2) ◽  
Author(s):  
Ján Jakubík
Keyword(s):  
Analogous Result ◽  
Radical Class ◽  
Partially Ordered ◽  
Ordered Groups ◽  
Lattice Ordered Groups ◽  
Mv Algebra ◽  
One To One ◽  
Abelian Lattice ◽  
Mv Algebras

AbstractFor an MV-algebra let J 0() be the system of all closed ideals of ; this system is partially ordered by the set-theoretical inclusion. A radical class X of MV-algebras will be called a K-radical class iff, whenever ∈ X and is an MV-algebra with J 0() ≅ J 0(), then ∈ X. An analogous notation for lattice ordered groups was introduced and studied by Conrad. In the present paper we show that there is a one-to-one correspondence between K-radical classes of MV-algebras and K-radical classes of abelian lattice ordered groups. We also prove an analogous result for product radical classes of MV-algebras; product radical classes of lattice ordered groups were studied by Ton.

Combining boolean algebras and ℓ-groups in the variety generated by chang’s mv-algebra

2016 ◽  
Vol 66 (2) ◽  
Author(s):  
A. Di Nola ◽  
A. R. Ferraioli ◽  
B. Gerla
Keyword(s):  
Boolean Algebra ◽  
Boolean Algebras ◽  
Maximal Ideals ◽  
Mv Algebra ◽  
Mv Algebras

AbstractIn this paper we investigate a class of MV-algebras built up by fixing a Boolean algebra, one of its maximal ideals and an

scholarly journals Rényi Entropy and Rényi Divergence in Product MV-Algebras

Entropy ◽  
2018 ◽  
Vol 20 (8) ◽  
pp. 587 ◽  
Author(s):  
Dagmar Markechová ◽  
Beloslav Riečan
Keyword(s):  
Shannon Entropy ◽  
Renyi Entropy ◽  
Rényi Entropy ◽  
Rényi Divergence ◽  
New Concepts ◽  
Leibler Divergence ◽  
Mv Algebra ◽  
The Relationship ◽  
Mv Algebras

This article deals with new concepts in a product MV-algebra, namely, with the concepts of Rényi entropy and Rényi divergence. We define the Rényi entropy of order q of a partition in a product MV-algebra and its conditional version and we study their properties. It is shown that the proposed concepts are consistent, in the case of the limit of q going to 1, with the Shannon entropy of partitions in a product MV-algebra defined and studied by Petrovičová (Soft Comput.2000, 4, 41–44). Moreover, we introduce and study the notion of Rényi divergence in a product MV-algebra. It is proven that the Kullback–Leibler divergence of states on a given product MV-algebra introduced by Markechová and Riečan in (Entropy2017, 19, 267) can be obtained as the limit of their Rényi divergence. In addition, the relationship between the Rényi entropy and the Rényi divergence as well as the relationship between the Rényi divergence and Kullback–Leibler divergence in a product MV-algebra are examined.

scholarly journals Rough Approximation Operators on a Complete Orthomodular Lattice

Axioms ◽  
2021 ◽  
Vol 10 (3) ◽  
pp. 164
Author(s):  
Songsong Dai
Keyword(s):  
Boolean Algebra ◽  
Orthomodular Lattice ◽  
Orthomodular Lattices ◽  
Approximation Operators ◽  
Rough Approximation ◽  
Distributive Law ◽  
Complete Orthomodular Lattice ◽  
The Relationship

This paper studies rough approximation via join and meet on a complete orthomodular lattice. Different from Boolean algebra, the distributive law of join over meet does not hold in orthomodular lattices. Some properties of rough approximation rely on the distributive law. Furthermore, we study the relationship among the distributive law, rough approximation and orthomodular lattice-valued relation.

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