The Euler characteristic and valuations on MV-algebras

2014 ◽  
Vol 64 (3) ◽  
Author(s):  
Daniele Mundici ◽  
Andrea Pedrini
Keyword(s):  
Euler Characteristic ◽  
Basic Element ◽  
Open Set ◽  
Maximal Ideals ◽  
Mv Algebra ◽  
Finitely Presented ◽  
Mv Algebras

AbstractEvery finitely presented MV-algebra A has a unique idempotent valuation E assigning value 1 to every basic element of A. For each a ∈ A, E(a) turns out to coincide with the Euler characteristic of the open set of maximal ideals m of A such that a/m is nonzero.

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

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

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 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 $$g(0) = 1_{Q^0 } $$ .

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 Some remarks on divisible polyhedral MV-algebras

2020 ◽  
Vol 70 (1) ◽  
pp. 51-60
Author(s):  
Serafina Lapenta
Keyword(s):  
Amalgamation Property ◽  
Finite Products ◽  
Finitely Presented ◽  
Mv Algebras

AbstractBuilding on similar notions for MV-algebras, polyhedral DMV-algebras are defined and investigated. For such algebras dualities with suitable categories of polyhedra are established, and the relation with finitely presented Riesz MV-algebras is investigated. Via hull-functors, finite products are interpreted in terms of hom-functors, and categories of polyhedral MV-algebras, polyhedral DMV-algebras and finitely presented Riesz MV-algebras are linked together. Moreover, the amalgamation property is proved for finitely presented DMV-algebras and Riesz MV-algebras, and for polyhedral DMV-algebras.

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.

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