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A Cantor-Bernstein theorem for σ -complete MV-algebras

Anna de Simone, Daniele Mundici, Mirko Navara (2003)

Czechoslovak Mathematical Journal

The Cantor-Bernstein theorem was extended to σ -complete boolean algebras by Sikorski and Tarski. Chang’s MV-algebras are a nontrivial generalization of boolean algebras: they stand to the infinite-valued calculus of Łukasiewicz as boolean algebras stand to the classical two-valued calculus. In this paper we further generalize the Cantor-Bernstein theorem to σ -complete MV-algebras, and compare it to a related result proved by Jakubík for certain complete MV-algebras.

A characterization of commutative basic algebras

Ivan Chajda (2009)

Mathematica Bohemica

A basic algebra is an algebra of the same type as an MV-algebra and it is in a one-to-one correspondence to a bounded lattice having antitone involutions on its principal filters. We present a simple criterion for checking whether a basic algebra is commutative or even an MV-algebra.

A glimpse of deductive systems in algebra

Dumitru Buşneag, Sergiu Rudeanu (2010)

Open Mathematics

The concept of a deductive system has been intensively studied in algebraic logic, per se and in connection with various types of filters. In this paper we introduce an axiomatization which shows how several resembling theorems that had been separately proved for various algebras of logic can be given unique proofs within this axiomatic framework. We thus recapture theorems already known in the literature, as well as new ones. As a by-product we introduce the class of pre-BCK algebras.

A non commutative generalization of -autonomous lattices

P. Emanovský, Jiří Rachůnek (2008)

Czechoslovak Mathematical Journal

Pseudo -autonomous lattices are non-commutative generalizations of -autonomous lattices. It is proved that the class of pseudo -autonomous lattices is a variety of algebras which is term equivalent to the class of dualizing residuated lattices. It is shown that the kernels of congruences of pseudo -autonomous lattices can be described as their normal ideals.

Additive closure operators on abelian unital l -groups

Filip Švrček (2006)

Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica

In the paper an additive closure operator on an abelian unital l -group ( G , u ) is introduced and one studies the mutual relation of such operators and of additive closure ones on the M V -algebra Γ ( G , u ) .

An atomic MV-effect algebra with non-atomic center

Vladimír Olejček (2007)


Does there exist an atomic lattice effect algebra with non-atomic subalgebra of sharp elements? An affirmative answer to this question (and slightly more) is given: An example of an atomic MV-effect algebra with a non-atomic Boolean subalgebra of sharp or central elements is presented.

Approximation Spacesin Non-commutative Generalizations of M V -algebras


Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica

Generalized MV-algebras (= GMV-algebras) are non-commutative generalizations of MV-algebras. They are an algebraic counterpart of the non-commutative Łukasiewicz infinite valued fuzzy logic. The paper investigates approximation spaces in GMV-algebras based on their normal ideals.

Archimedean atomic lattice effect algebras in which all sharp elements are central

Zdena Riečanová (2006)


We prove that every Archimedean atomic lattice effect algebra the center of which coincides with the set of all sharp elements is isomorphic to a subdirect product of horizontal sums of finite chains, and conversely. We show that every such effect algebra can be densely embedded into a complete effect algebra (its MacNeille completion) and that there exists an order continuous state on it.

Archimedean GMV-chains

Jan Kühr (2002)

Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica

Atomicity of lattice effect algebras and their sub-lattice effect algebras

Jan Paseka, Zdena Riečanová (2009)


We show some families of lattice effect algebras (a common generalization of orthomodular lattices and MV-effect algebras) each element E of which has atomic center C(E) or the subset S(E) of all sharp elements, resp. the center of compatibility B(E) or every block M of E. The atomicity of E or its sub-lattice effect algebras C(E), S(E), B(E) and blocks M of E is very useful equipment for the investigations of its algebraic and topological properties, the existence or smearing of states on E, questions...

Banaschewski’s theorem for generalized M V -algebras

Ján Jakubík (2007)

Czechoslovak Mathematical Journal

A generalized M V -algebra 𝒜 is called representable if it is a subdirect product of linearly ordered generalized M V -algebras. Let S be the system of all congruence relations ρ on 𝒜 such that the quotient algebra 𝒜 / ρ is representable. In the present paper we prove that the system S has a least element.

Basic pseudorings

Ivan Chajda, Miroslav Kolařík (2009)

Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica

The concept of a basic pseudoring is introduced. It is shown that every orthomodular lattice can be converted into a basic pseudoring by using of the term operation called Sasaki projection. It is given a mutual relationship between basic algebras and basic pseudorings. There are characterized basic pseudorings which can be converted into othomodular lattices.

Bipartite pseudo MV-algebras

Grzegorz Dymek (2006)

Discussiones Mathematicae - General Algebra and Applications

A bipartite pseudo MV-algebra A is a pseudo MV-algebra such that A = M ∪ M ̃ for some proper ideal M of A. This class of pseudo MV-algebras, denoted BP, is investigated. The class of pseudo MV-algebras A such that A = M ∪ M ̃ for all maximal ideals M of A, denoted BP₀, is also studied and characterized.

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