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.

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.

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.

Radical of a-Ideals in Mv -Modules

2014 ◽  
Vol 0 (0) ◽  
Author(s):  
F. Forouzesh ◽  
E. Eslami ◽  
A. Borumand Saeid
Keyword(s):  
Maximal Ideal ◽  
The Other ◽  
Subject Classification ◽  
Mv Algebra ◽  
Mv Algebras

Abstract In this paper, we introduce the notion of the radical of an ideal in MV - algebras. Several characterizations of this radical is given. We define the notion of a semi-maximal ideal in an MV -algebra and prove some theorems which give relations between this semi-maximal ideal and the other types of ideals in MV -algebras. Also we prove that A/I is a semi-simple MV -algebra if and only if I is a semi-maximal ideal of an MV -algebra A. The above notions are used to define the radical of A-ideals in MV -modules and investigate some properties. Mathematics Subject Classification 2010: 03B50, 03G25, 06D35

scholarly journals Residuation in non-associative MV-algebras

2018 ◽  
Vol 68 (6) ◽  
pp. 1313-1320
Author(s):  
Ivan Chajda ◽  
Helmut Länger
Keyword(s):  
Binary Operation ◽  
Residuated Lattice ◽  
Natural Question ◽  
Natural Conditions ◽  
Double Negation ◽  
Mv Algebra ◽  
Residuated Poset ◽  
Double Negation Law ◽  
Mv Algebras

Abstract It is well known that every MV-algebra can be converted into a residuated lattice satisfying divisibility and the double negation law. In a previous paper the first author and J. Kühr introduced the concept of an NMV-algebra which is a non-associative modification of an MV-algebra. The natural question arises if an NMV-algebra can be converted into a residuated structure, too. Contrary to MV-algebras, NMV-algebras are not based on lattices but only on directed posets and the binary operation need not be associative and hence we cannot expect to obtain a residuated lattice but only an essentially weaker structure called a conditionally residuated poset. Considering several additional natural conditions we show that every NMV-algebra can be converted in such a structure. Also conversely, every such structure can be organized into an NMV-algebra. Further, we study an a bit more stronger version of an algebra where the binary operation is even monotone. We show that such an algebra can be organized into a residuated poset and, conversely, every residuated poset can be converted in this structure.

scholarly journals Isometries and direct decompositions of pseudo MV-algebras

2007 ◽  
Vol 57 (2) ◽  
Author(s):  
Milan Jasem
Keyword(s):  
Direct Decomposition ◽  
The Other ◽  
Other Hand ◽  
Mv Algebra ◽  
Direct Decompositions ◽  
Mv Algebras

AbstractIn the paper isometries in pseudo MV-algebras are investigated. It is shown that for every isometry f in a pseudo MV-algebra $$\mathcal{A}$$ = (A, ⊕, −, ∼, 0, 1) there exists an internal direct decomposition $$\mathcal{A} = \mathcal{B}^0 \times \mathcal{C}^0 $$ of $$\mathcal{A}$$ with $$\mathcal{C}^0 $$ commutative such that $$f(0) = 1_{C^0 } $$ and $$f(x) = x_{B^0 } \oplus (1_{C^0 } \odot (x_{C^0 } )^ - ) = x_{B^0 } \oplus (1_{C^0 } - x_{C^0 } )$$ for each x ∈ A.On the other hand, if $$\mathcal{A} = \mathcal{P}^0 \times \mathcal{Q}^0 $$ is an internal direct decomposition of a pseudo MV-algebra $$\mathcal{A}$$ = (A, ⊕, −, ∼, 0, 1) with $$\mathcal{Q}^0 $$ commutative, then the mapping g: A → A defined by $$g(x) = x_{P^0 } \oplus (1_{Q^0 } - x_{Q^0 } )$$ is an isometry in $$\mathcal{A}$$ and $$g(0) = 1_{Q^0 } $$ .

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.

Higher degrees of distributivity in complete generalized MV-algebras

2011 ◽  
Vol 61 (3) ◽  
Author(s):  
Ján Jakubík
Keyword(s):  
Mv Algebra ◽  
Mv Algebras

AbstractWe apply the notion of generalized MV-algebra (GMV-algebra, in short) in the sense of Galatos and Tsinakis. Let M be a complete GMV-algebra and let α be a cardinal. We prove that M is α-distributive if and only if it is (α, 2)-distributive. We deal with direct summands of M which are homogeneous with respect to higher degrees of distributivity.

scholarly journals Pseudo MV-algebras are intervals in ℓ-groups

2002 ◽  
Vol 72 (3) ◽  
pp. 427-446 ◽  
Author(s):  
Anatolij Dvurečenskij
Keyword(s):  
Strong Unit ◽  
Famous Result ◽  
Mv Algebra ◽  
Mv Algebras

AbstractWe show that any pseudo MV-algebra is isomorphic with an interval Γ(G, u), where G is an ℓ-group not necessarily Abelian with a strong unit u. In addition, we prove that the category of unital ℓ-groups is categorically equivalent with the category of pseudo MV-algebras. Since pseudo MV-algebras are a non-commutative generalization of MV-algebras, our assertions generalize a famous result of Mundici for a representation of MV-algebras by Abelian unital ℓ-groups. Our methods are completely different from those of Mundici. In addition, we show that any Archimedean pseudo MV-algebra is an MV-algebra.

On idempotent modifications of generalized MV-algebras

2010 ◽  
Vol 60 (2) ◽  
Author(s):  
Ján Jakubík
Keyword(s):  
Present Note ◽  
Subdirectly Irreducible ◽  
Mv Algebra ◽  
Mv Algebras

AbstractThe notion of idempotent modification of an algebra was introduced by Ježek; he proved that the idempotent modification of a group is always subdirectly irreducible. In the present note we show that the idempotent modification of a generalized MV -algebra having more than two elements is directly irreducible if and only if there exists an element in A which fails to be boolean. Some further results on idempotent modifications are also proved.

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