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Direct product decompositions of bounded commutative residuated ℓ -monoids

Ján Jakubík (2008)

Czechoslovak Mathematical Journal

The notion of bounded commutative residuated ℓ -monoid ( B C R ℓ -monoid, in short) generalizes both the notions of M V -algebra and of B L -algebra. Let A ̧ be a B C R ℓ -monoid; we denote by ℓ ( A ̧ ) the underlying lattice of A ̧ . In the present paper we show that each direct...

Direct summands and retract mappings of generalized M V -algebras

Ján Jakubík (2008)

Czechoslovak Mathematical Journal

In the present paper we deal with generalized M V -algebras ( G M V -algebras, in short) in the sense of Galatos and Tsinakis. According to a result of the mentioned authors, G M V -algebras can be obtained by a truncation construction from lattice ordered groups. We investigate direct summands and retract mappings of G M V -algebras. The relations between G M V -algebras and lattice ordered groups are essential for this investigation.

Directoids with sectionally antitone involutions and skew MV-algebras

Ivan Chajda, Miroslav Kolařík (2007)

Mathematica Bohemica

It is well-known that every MV-algebra is a distributive lattice with respect to the induced order. Replacing this lattice by the so-called directoid (introduced by J. Ježek and R. Quackenbush) we obtain a weaker structure, the so-called skew MV-algebra. The paper is devoted to the axiomatization of skew MV-algebras, their properties and a description of the induced implication algebras.

Distributivity of bounded lattices with sectionally antitone involutions

Ivan Chajda (2005)

Discussiones Mathematicae - General Algebra and Applications

We present a simple condition under which a bounded lattice L with sectionally antitone involutions becomes an MV-algebra. In thiscase, L is distributive. However, we get a criterion characterizingdistributivity of L in terms of antitone involutions only.

Embeddings of totally ordered MV-algebras of bounded cardinality

Piotr J. Wojciechowski (2009)

Fundamenta Mathematicae

For a given cardinal number 𝔞, we construct a totally ordered MV-algebra M(𝔞) having the property that every totally ordered MV-algebra of cardinality at most 𝔞 embeds into M(𝔞). In case 𝔞 = ℵ₀, the algebra M(𝔞) is the first known MV-algebra with respect to which the deductive system for the infinitely-valued Łukasiewicz's propositional logic is strongly complete.

Entropy on effect algebras with the Riesz decomposition property I: Basic properties

Antonio Di Nola, Anatolij Dvurečenskij, Marek Hyčko, Corrado Manara (2005)

Kybernetika

We define the entropy, lower and upper entropy, and the conditional entropy of a dynamical system consisting of an effect algebra with the Riesz decomposition property, a state, and a transformation. Such effect algebras allow many refinements of two partitions. We present the basic properties of these entropies and these notions are illustrated by many examples. Entropy on MV-algebras is postponed to Part II.

Epimorphisms between finite MV-algebras

Aldo V. Figallo, Marina B. Lattanzi (2017)

Mathematica Bohemica

MV-algebras were introduced by Chang to prove the completeness of the infinite-valued Łukasiewicz propositional calculus. Recently, algebraic theory of MV-algebras has been intensively studied. Wajsberg algebras are just a reformulation of Chang MV-algebras where implication is used instead of disjunction. Using these equivalence, in this paper we provide conditions for the existence of an epimorphism between two finite MV-algebras A and B . Specifically, we define the mv-functions with domain in...

Extensional subobjects in categories of Ω -fuzzy sets

Jiří Močkoř (2007)

Czechoslovak Mathematical Journal

Two categories 𝕊𝕖𝕥 ( Ω ) and 𝕊𝕖𝕥𝔽 ( Ω ) of fuzzy sets over an M V -algebra Ω are investigated. Full subcategories of these categories are introduced consisting of objects ( s u b ( A , δ ) , σ ) , where s u b ( A , δ ) is a subset of all extensional subobjects of an object ( A , δ ) . It is proved that all these subcategories are quasi-reflective subcategories in the corresponding categories.

Folding theory applied to BL-algebras

Young Jun, Jung Ko (2004)

Open Mathematics

The notion of n-fold grisly deductive systems is introduced. Some conditions for a deductive system to be an n-fold grisly deductive system are provided. Extension property for n-fold grisly deductive system is established.

Fuzzy n-fold integral filters in BL-algebras

Rajab Ali Borzooei, Akbar Paad (2013)

Discussiones Mathematicae - General Algebra and Applications

In this paper, we introduce the notion of fuzzy n-fold integral filter in BL-algebras and we state and prove several properties of fuzzy n-fold integral filters. Using a level subset of a fuzzy set in a BL-algebra, we give a characterization of fuzzy n-fold integral filters. Also, we prove that the homomorphic image and preimage of fuzzy n-fold integral filters are also fuzzy n-fold integral filters. Finally, we study the relationship among fuzzy n-fold obstinate filters, fuzzy n-fold integral filters...

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