8 results
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2. Prime, minimal prime and maximal ideals spaces in residuated lattices.
- Author
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Piciu, Dana
- Subjects
- *
PRIME ideals , *RESIDUATED lattices , *TOPOLOGICAL spaces , *TOPOLOGY - Abstract
In this paper, the notion of minimal prime ideal is introduced in residuated lattices and related properties are investigated. Also, new equivalent characterizations and properties for prime and maximal ideals are obtained and the relation between these ideals and minimal prime ideals is discussed for De Morgan residuated lattices. Moreover, we prove that it is possible to introduce and study, by a standard way, Zariski topology on the lattice P (L) of prime ideals of any residuated lattice L. Also, since m P (L) , the set of minimal prime ideals of L , and M (L) , the set of maximal ideals of L , are subsets of P (L) , we endow m P (L) and M (L) with the topology induced by the Zariski topology on P (L) and we characterize these topological spaces for residuated lattices. [ABSTRACT FROM AUTHOR]
- Published
- 2021
- Full Text
- View/download PDF
3. On EMV-algebras.
- Author
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Dvurečenskij, Anatolij and Zahiri, Omid
- Subjects
- *
FUZZY sets , *BOOLEAN algebra , *ABELIAN categories , *SET functions , *IDEMPOTENTS , *CONGRUENCE lattices , *IDEALS (Algebra) , *SEMISIMPLE Lie groups - Abstract
The paper deals with an algebraic extension of MV-algebras based on the definition of generalized Boolean algebras. We introduce a new class of structures, not necessarily with a top element, which are called EMV-algebras, in a way that every EMV-algebra contains an MV-algebra. First, we present basic properties of EMV-algebras. We give some examples, introduce and investigate congruence relations, ideals and filters on these algebras. We establish a basic representation result saying that each EMV-algebra can be embedded into an EMV-algebra with top element and we characterize EMV-algebras either as structures which are termwise equivalent to MV-algebras or as maximal ideals of EMV-algebras with top element. We study the lattice of ideals of an EMV-algebra and prove that every EMV-algebra has at least one maximal ideal. We define an EMV-clan of fuzzy sets as a special EMV-algebra where all operations are defined by points. We show that any semisimple EMV-algebra is isomorphic to an EMV-clan of fuzzy functions on a set. The set of EMV-algebras is neither a variety nor a quasivariety, but rather a special class of EMV-algebras which we call a q-variety of EMV-algebras. We present an equational base for each proper q-subvariety of the q-variety of EMV-algebras. We establish categorical equivalencies among the category of proper EMV-algebras, the category of MV-algebras with a fixed special maximal ideal, and a special category of Abelian unital ℓ -groups. [ABSTRACT FROM AUTHOR]
- Published
- 2019
- Full Text
- View/download PDF
4. On (para, quasi) topological MV-algebras.
- Author
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Najafi, Marziyeh, Rezaei, Gholam Reza, and Kouhestani, Nader
- Subjects
- *
TOPOLOGICAL algebras , *TOPOLOGICAL semigroups , *VECTOR topology , *ALGEBRAIC topology , *ALGORITHMS - Abstract
In this paper, the notions of (para, quasi, semi) topological MV-algebras are defined and their related properties are studied. Also, topologies with which an MV-algebra can be a (para, semi) topological MV-algebra are obtained. Clearly, a topological MV-algebra is a (para, quasi, semi) topological MV-algebra, but the converse is not true, as shown by an example. In addition, we study ideals and filters in (para, quasi) topological MV-algebras, and we show that a quasitopological MV-algebra is a topological MV-algebra if the ideal {0}, or equivalently, the filter {1} is open. [ABSTRACT FROM AUTHOR]
- Published
- 2017
- Full Text
- View/download PDF
5. Ideals and involutive filters in generalizations of fuzzy structures.
- Author
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Rachůnek, Jiří and Šalounová, Dana
- Subjects
- *
IDEALS (Algebra) , *FILTERS (Mathematics) , *FUZZY logic , *INTEGRALS , *LATTICE theory , *MANY-valued logic - Abstract
(Bounded integral) residuated lattices (which need not be commutative) form a large class of algebras containing some classes of algebras behind many-valued and fuzzy logics. Congruences of such algebras are usually defined and investigated by means of their normal filters. In the paper we introduce and investigate ideals of residuated lattices. We show that one can define, in some cases, congruences also using ideals and that the corresponding quotient residuated lattices are involutive. [ABSTRACT FROM AUTHOR]
- Published
- 2017
- Full Text
- View/download PDF
6. Categorical foundations of variety-based bornology.
- Author
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Paseka, Jan and Solovyov, Sergey A.
- Subjects
- *
ISOMORPHISM (Mathematics) , *SET theory , *BORNOLOGICAL spaces , *MATHEMATICAL analysis , *LATTICE field theory - Abstract
Following the concept of topological theory of S.E. Rodabaugh, this paper introduces a new approach to (lattice-valued) bornology, which is based in bornological theories, and which is called variety-based bornology. In particular, motivated by the notion of topological system of S. Vickers, we introduce the concept of variety-based bornological system, and show that the category of variety-based bornological spaces is isomorphic to a full reflective subcategory of the category of variety-based bornological systems. [ABSTRACT FROM AUTHOR]
- Published
- 2016
- Full Text
- View/download PDF
7. States on commutative basic algebras
- Author
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Botur, Michal, Halaš, Radomír, and Kühr, Jan
- Subjects
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COMMUTATIVE algebra , *GENERALIZATION , *FUZZY logic , *NONASSOCIATIVE algebras , *MATHEMATICAL functions , *MATHEMATICAL proofs - Abstract
Abstract: The paper deals with states on commutative basic algebras that are a non-associative generalization of MV-algebras or, in other words, the algebraic semantics for a fuzzy logic which generalizes the Łukasiewicz logic in that the conjunction is not associative. States are defined in the same way as Mundici''s states on MV-algebras as normalized finitely additive [0,1]-valued functions, and some results analogous to the results that are known from MV-algebras are proved. [Copyright &y& Elsevier]
- Published
- 2012
- Full Text
- View/download PDF
8. Ideal-valued topological structures
- Author
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Gutiérrez García, J., Kubiak, T., and Šostak, A.P.
- Subjects
- *
FUZZY topology , *CONTINUOUS lattices , *ADJUNCTION theory , *TOPOLOGICAL spaces , *FUNCTOR theory , *IDEALS (Algebra) - Abstract
Abstract: With L a complete lattice and M a continuous lattice, this paper demonstrates an adjunction between M -valued L-topological spaces (i.e. (L,M )-topological spaces) and Idl(M )-valued L-topological spaces where Idl(M ) is the complete lattice of all ideals of M . It is shown that the right adjoint functor provides a procedure of generating (L,M )-topologies from antitone families of (L,M )-topologies. This procedure is then applied to give an internal characterization of joins in the complete lattice of all (L,M )-topologies on a given set. [Copyright &y& Elsevier]
- Published
- 2010
- Full Text
- View/download PDF
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