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Anatolij Dvurečenskij - One of the best experts on this subject based on the ideXlab platform.
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States on wEMV-algebras
Bollettino dell'Unione Matematica Italiana, 2020Co-Authors: Anatolij DvurečenskijAbstract:Recently in Dvurečenskij and Zahiri (A variety containing EMV-algebras and Pierce sheaves, arXiv:1911.06625 ), new algebras called wEMV-algebras, which generalize MV-algebras, generalized Boolean algebras and EMV-algebras, were founded, and for these algebras a top element is not assumed a priori. For this class we define a state as a mapping from a wEMV-algebra into the real interval [0, 1] which preserves a kind of subtraction of two comparable elements and attaining the value 1 in some element. It can happen that some wEMV-algebras are stateless, e.g. cancellative ones. We characterize extremal states just as state-morphisms which are wEMV-homomorphisms from an algebra into the real interval [0, 1]. We show that there is a one-to-one correspondence between the set of state-morphisms and the set of maximal ideals having a special property. Moreover, we prove that under some conditions every state on a wEMV-algebra is a weak limit of a net of convex combinations of state-morphisms.
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Observables on perfect MV-algebras
Fuzzy Sets and Systems, 2019Co-Authors: Antonio Di Nola, Anatolij Dvurečenskij, Giacomo LenziAbstract:Abstract An observable on an MV-algebra is any σ-homomorphism from the Borel σ-algebra B ( R ) into the MV-algebra which maps a sequence of disjoint Borel sets onto summable elements of the MV-algebra. We establish that there is a one-to-one correspondence between observables on Rad -Dedekind σ-complete perfect MV-algebras with principal radicals and their spectral resolutions. It means that we show that our partial information on an observable known only on all intervals of the form ( − ∞ , t ) is sufficient to determine the whole information about the observable. In addition, this correspondence allows us to define the Olson order which is a partial order on the set O ( M ) of all observables on an MV-algebra M as well as, we can define a sum of observables, so that O ( M ) becomes a lattice-ordered semigroup.
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Pseudo MV-algebras and lexicographic product☆
Fuzzy Sets and Systems, 2016Co-Authors: Anatolij DvurečenskijAbstract:Abstract We study algebraic conditions when a pseudo MV-algebra is an interval in the lexicographic product of an Abelian unital linearly ordered group and an l-group that is not necessarily Abelian. We introduce two classes of pseudo MV-algebras which can be split into a system of comparable slices indexed by elements of an interval in an Abelian linearly ordered group. We show when such pseudo MV-algebras have a representation by a lexicographic product with an l-group. Fixing a unital l-group, we show that the category of such pseudo MV-algebras is categorically equivalent to the category of l-groups.
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Orthocomplete Pseudo MV -algebras
International Journal of General Systems, 2016Co-Authors: Anatolij Dvurečenskij, Omid ZahiriAbstract:Pseudo MV-algebras are a non-commutative generalization of MV-algebras. The main purpose of the paper is to introduce and investigate orthocomplete pseudo MV-algebras. We use the concepts of projectable pseudo MV-algebras and large pseudo MV-subalgebras to introduce orthocomplete pseudo MV-algebras. Then we apply a generalization of the Mundici functor to an orthocompletion of an representable l-group to prove that each representable pseudo MV-algebra has an orthocompletion. In particular, our results are valid also for MV-algebras.
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Lexicographic pseudo MV-algebras
Journal of Applied Logic, 2015Co-Authors: Anatolij DvurečenskijAbstract:A lexicographic pseudo MV-algebra is an algebra that is isomorphic to an interval in the lexicographic product of a unital linearly ordered group with an arbitrary ?-group. We present conditions when a pseudo MV-algebra is lexicographic. We show that a key condition is the existence of a lexicographic ideal, or equivalently, a case when the algebra can be split into comparable slices indexed by elements of the interval 0 , u of some unital linearly ordered group ( H , u ) . Finally, we show that fixing ( H , u ) , the category of ( H , u ) -lexicographic pseudo MV-algebras is categorically equivalent to the category of ?-groups.
Antonio Di Nola - One of the best experts on this subject based on the ideXlab platform.
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On MV-algebras of non-linear functions
2020Co-Authors: Antonio Di Nola, Giacomo Lenzi, Gaetano VitaleAbstract:In this paper, the main results are:a study of the finitely generated MV-algebras of continuous functions from the n-th power of the unit real interval I to I;a study of Hopfian MV-algebras; anda category-theoretic study of the map sending an MV-algebra as above to the range of its generators (up to a suitable form of homeomorphism).
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On the variety of Gödel MV-algebras
Soft Computing, 2019Co-Authors: Antonio Di Nola, Revaz Grigolia, Gaetano VitaleAbstract:We introduce a new algebraic structure $$\begin{aligned} (A, \otimes , \oplus , *, \vee , \wedge , \rightharpoonup , 0, 1) \end{aligned}$$ ( A , ⊗ , ⊕ , ∗ , ∨ , ∧ , ⇀ , 0 , 1 ) called Gödel – MV-algebra ( GMV - algebra ) such that $$(A, \otimes , \oplus , *, 0, 1)$$ ( A , ⊗ , ⊕ , ∗ , 0 , 1 ) is MV -algebra; $$(A,\vee , \wedge ,\rightharpoonup , 0, 1)$$ ( A , ∨ , ∧ , ⇀ , 0 , 1 ) is a Gödel algebra (i. e. Heyting algebra satisfying the identity $$(x \rightharpoonup y ) \vee (y \rightharpoonup x ) =1$$ ( x ⇀ y ) ∨ ( y ⇀ x ) = 1 ). It is shown that the lattice of congruences of a GMV -algebra $$(A, \otimes , \oplus , *, \rightharpoonup , 0, 1)$$ ( A , ⊗ , ⊕ , ∗ , ⇀ , 0 , 1 ) is isomorphic to the lattice of Skolem filters (i. e. special type of MV -filters) of the MV -algebra $$(A, \otimes , \oplus , *, 0, 1)$$ ( A , ⊗ , ⊕ , ∗ , 0 , 1 ) . Any GMV -algebra is bi-Heyting algebra. Any chain GMV -algebra is simple, and any GMV -algebra is semi-simple. Finitely generated GMV -algebras are described, and finitely generated finitely presented GMV -algebras are characterized. The algebraic counterpart of axiomatically presented GMV -logic is GMV -algebras .
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Observables on perfect MV-algebras
Fuzzy Sets and Systems, 2019Co-Authors: Antonio Di Nola, Anatolij Dvurečenskij, Giacomo LenziAbstract:Abstract An observable on an MV-algebra is any σ-homomorphism from the Borel σ-algebra B ( R ) into the MV-algebra which maps a sequence of disjoint Borel sets onto summable elements of the MV-algebra. We establish that there is a one-to-one correspondence between observables on Rad -Dedekind σ-complete perfect MV-algebras with principal radicals and their spectral resolutions. It means that we show that our partial information on an observable known only on all intervals of the form ( − ∞ , t ) is sufficient to determine the whole information about the observable. In addition, this correspondence allows us to define the Olson order which is a partial order on the set O ( M ) of all observables on an MV-algebra M as well as, we can define a sum of observables, so that O ( M ) becomes a lattice-ordered semigroup.
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Topological spaces of monadic MV-algebras
Soft Computing, 2019Co-Authors: Antonio Di Nola, Revaz Grigolia, Giacomo LenziAbstract:We construct a covariant functor $$\gamma $$ γ from the category of monadic MV -algebras into the category of Q -distributive lattices, i.e., distributive lattices with quantifier introduced by R. Cignoli. For every monadic MV -algebra, we construct a dual object named QM -space; these objects form a special subcategory of spectral spaces and of Q -spaces developed by R. Cignoli for Q -distributive lattices.
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Relative subalgebras of MV-algebras
Algebra Universalis, 2017Co-Authors: L. P. Belluce, Antonio Di Nola, Giacomo LenziAbstract:Given an MV-algebra A, with its natural partial ordering, we consider in A the intervals of the form [0, a], where \({a \in A}\). These intervals have a natural structure of MV-algebras and will be called the relative subalgebras of A (in analogy with Boolean algebras). We investigate various properties of relative subalgebras and their relations with the original MV-algebra.
Omid Zahiri - One of the best experts on this subject based on the ideXlab platform.
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Orthocomplete Pseudo MV -algebras
International Journal of General Systems, 2016Co-Authors: Anatolij Dvurečenskij, Omid ZahiriAbstract:Pseudo MV-algebras are a non-commutative generalization of MV-algebras. The main purpose of the paper is to introduce and investigate orthocomplete pseudo MV-algebras. We use the concepts of projectable pseudo MV-algebras and large pseudo MV-subalgebras to introduce orthocomplete pseudo MV-algebras. Then we apply a generalization of the Mundici functor to an orthocompletion of an representable l-group to prove that each representable pseudo MV-algebra has an orthocompletion. In particular, our results are valid also for MV-algebras.
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Orthocomplete Pseudo MV-algebras
arXiv: Rings and Algebras, 2015Co-Authors: Anatolij Dvurečenskij, Omid ZahiriAbstract:Pseudo $MV$-algebras are a non-commutative generalization of $MV$-algebras. The main purpose of the paper is to introduce and investigate orthocomplete pseudo $MV$-algebras. We use the concepts of projectable pseudo $MV$-algebras and large pseudo $MV$-subalgebras to introduce orthocomplete pseudo $MV$-algebras. Then we apply a generalization of the Mundici's functor to an orthocompletion of an representable $\ell$-group to prove that each representable pseudo $MV$-algebra has an orthocompletion. In particular, our results are valid also for $MV$-algebras.
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Some remarks on hyper MV-algebras
Journal of Intelligent and Fuzzy Systems, 2014Co-Authors: Rajab Ali Borzooei, Wieslaw A. Dudek, A. Radfar, Omid ZahiriAbstract:We describe relations between hyper MV-algebras and hyper K-algebras and prove that a finite hyper MV-algebra satisfying the condition $0\CirclePlus x = \{x\}$, used by many authors, is in fact an ordinary MV-algebra. We also characterize relations between the main types of deductive systems of hyper MV-algebras.
Giacomo Lenzi - One of the best experts on this subject based on the ideXlab platform.
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On MV-algebras of non-linear functions
2020Co-Authors: Antonio Di Nola, Giacomo Lenzi, Gaetano VitaleAbstract:In this paper, the main results are:a study of the finitely generated MV-algebras of continuous functions from the n-th power of the unit real interval I to I;a study of Hopfian MV-algebras; anda category-theoretic study of the map sending an MV-algebra as above to the range of its generators (up to a suitable form of homeomorphism).
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A characterization of pseudofinite MV-algebras
Soft Computing, 2020Co-Authors: Eslam Farsimadan, Giacomo Lenzi, Paolo Rizzo, Arsham Borumand SaeidAbstract:We consider pseudofinite MV-algebras. As a main result, we show that an infinite MV-algebra is pseudofinite if and only if it is definably well founded, improving a result of a previous paper. Moreover, we show that the theory of pseudofinite MV-algebras has a partial form of elimination of quantifiers. Further, we show that the class of pseudofinite MV-chains and the class of pseudofinite MV-algebras are not finitely axiomatizable, we give some collapsing results for pseudofinite MV-algebras, we consider relative subalgebras of pseudofinite MV-algebras, and we study ideals of pseudofinite MV-algebras.
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Observables on perfect MV-algebras
Fuzzy Sets and Systems, 2019Co-Authors: Antonio Di Nola, Anatolij Dvurečenskij, Giacomo LenziAbstract:Abstract An observable on an MV-algebra is any σ-homomorphism from the Borel σ-algebra B ( R ) into the MV-algebra which maps a sequence of disjoint Borel sets onto summable elements of the MV-algebra. We establish that there is a one-to-one correspondence between observables on Rad -Dedekind σ-complete perfect MV-algebras with principal radicals and their spectral resolutions. It means that we show that our partial information on an observable known only on all intervals of the form ( − ∞ , t ) is sufficient to determine the whole information about the observable. In addition, this correspondence allows us to define the Olson order which is a partial order on the set O ( M ) of all observables on an MV-algebra M as well as, we can define a sum of observables, so that O ( M ) becomes a lattice-ordered semigroup.
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Topological spaces of monadic MV-algebras
Soft Computing, 2019Co-Authors: Antonio Di Nola, Revaz Grigolia, Giacomo LenziAbstract:We construct a covariant functor $$\gamma $$ γ from the category of monadic MV -algebras into the category of Q -distributive lattices, i.e., distributive lattices with quantifier introduced by R. Cignoli. For every monadic MV -algebra, we construct a dual object named QM -space; these objects form a special subcategory of spectral spaces and of Q -spaces developed by R. Cignoli for Q -distributive lattices.
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Relative subalgebras of MV-algebras
Algebra Universalis, 2017Co-Authors: L. P. Belluce, Antonio Di Nola, Giacomo LenziAbstract:Given an MV-algebra A, with its natural partial ordering, we consider in A the intervals of the form [0, a], where \({a \in A}\). These intervals have a natural structure of MV-algebras and will be called the relative subalgebras of A (in analogy with Boolean algebras). We investigate various properties of relative subalgebras and their relations with the original MV-algebra.
Jean B. Nganou - One of the best experts on this subject based on the ideXlab platform.
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Stone MV-algebras and strongly complete MV-algebras
Algebra universalis, 2017Co-Authors: Jean B. NganouAbstract:Characterizations of compact Hausdorff topological MV-algebras, Stone MV-algebras, and MV-algebras that are isomorphic to their profinite completions are established. It is proved that compact Hausdorff topological MV-algebras are products (both topological and algebraic) of copies [0, 1] with the interval topology and finite Łukasiewicz chains with the discrete topology. Going one step further, we also prove that Stone MV-algebras are products (both topological and algebraic) of finite Łukasiewicz chains with the discrete topology. Finally, it is proved that an MV-algebra is isomorphic to its profinite completion if and only if it is profinite and each of its maximal ideals of finite rank is principal.
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Profinite Completions of MV-algebras
arXiv: Logic, 2016Co-Authors: Jean B. NganouAbstract:We determine the profinite completions of MV-algebras, and obtain a description that generalizes the well known profinite completions of Boolean algebras as the power sets of their Stone spaces. We also use the description found to prove that every profinite MV-algebra is the profinite completion of a separating MV-algebra of $[0,1]$-valued continuous functions on some nonempty compact Hausdorff space, with pointwise operations.
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Stone MV-algebras and Strongly complete MV-algebras
arXiv: Logic, 2015Co-Authors: Jean B. NganouAbstract:Compact Hausdorff topological MV-algebras and Stone MV-algebras are completely characterized. We obtain that compact Hausdorff topological MV-algebras are product (both topological and algebraic) of copies $[0,1]$ with standard topology and finite Lukasiewicz chains with discrete topology. Going one step further we also prove that Stone MV-algebras are product (both topological and algebraic) of finite Lukasiewicz chains with discrete topology. We also prove that an MV-algebra is strongly complete (isomorphic to its profinite completion) if and only if it is profinite and its maximal ideals of finite ranks are principal.
