On the category Set(JCPos)

Solovyov, Sergey A.

doi:10.1016/j.fss.2005.06.007

Cited by 20 publications

(6 citation statements)

References 11 publications

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“…The approach of this paper (the definition of morphisms) in its turn relies on the category Set(CSLat( )) of lattice-valued sets of [49] (which is the category CSLat( )-Set in the notations of this paper), and seems to be more appropriate, since it fuzzifies not only algebras, but also their respective homomorphisms (which in category theory is even more important). Moreover, one can easily obtain the next result (recall our notation for constant maps, mentioned before Proposition 5).…”

Section: Definitionmentioning

confidence: 99%

Sobriety and spatiality in categories of lattice-valued algebras

Solovyov

2012

Fuzzy Sets and Systems

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Section: Definitionmentioning

confidence: 99%

Sobriety and spatiality in categories of lattice-valued algebras

Solovyov

2012

Fuzzy Sets and Systems

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“…We first consider the situation in a Goguen style category similar to that studied by Solovyov in [17] and then look at the categorical fabric I talked about in Linz in 2004.…”

Section: Organizing Fuzzy Predicate Logics Categoricallymentioning

confidence: 99%

A categorical semantics for fuzzy predicate logic

Stout

2010

Fuzzy Sets and Systems

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“…Definition 1.23 (cf. [37,1,[38][39][40][41][42][43][44][45][46][47][48][49][50]). Fixing L ∈ |SQuant|, the category L-FQTop has ground category Set and comprises the following data:…”

Section: Topological Preliminariesmentioning

confidence: 99%

Relationship of Algebraic Theories to Powerset Theories and Fuzzy Topological Theories for Lattice-Valued Mathematics

Rodabaugh

2007

International Journal of Mathematics and Mathematical Sciences

102

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This paper deals with a broad question—to what extent is topology algebraic—using two specific questions: (1) what are the algebraic conditions on the underlying membership lattices which insure that categories for topology and fuzzy topology are indeed topological categories; and (2) what are the algebraic conditions which insure that algebraic theories in the sense of Manes are a foundation for the powerset theories generating topological categories for topology and fuzzy topology? This paper answers the first question by generalizing the Höhle-Šostak foundations for fixed-basis lattice-valued topology and the Rodabaugh foundations for variable-basis lattice-valued topology using semi-quantales; and it answers the second question by giving necessary and sufficient conditions under which certain theories—the very ones generating powerset theories generating (fuzzy) topological theories in the sense of this paper—are algebraic theories, and these conditions use unital quantales. The algebraic conditions answering the second question are much stronger than those answering the first question. The syntactic benefits of having an algebraic theory as a foundation for the powerset theory underlying a (fuzzy) topological theory are explored; the relationship between these two specific questions is discussed; the role of pseudo-adjoints is identified in variable-basis powerset theories which are algebraically generated; the relationships between topological theories in the sense of Adámek-Herrlich-Strecker and topological theories in the sense of this paper are fully resolved; lower-image operators introduced for fixed-basis mathematics are completely described in terms of standard image operators; certain algebraic theories are given which determine powerset theories determining a new class of variable-basis categories for topology and fuzzy topology using new preimage operators; and the theories of this paper are undergirded throughout by several extensive inventories of examples.

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On the category Set(JCPos)

Cited by 20 publications

References 11 publications

Sobriety and spatiality in categories of lattice-valued algebras

Sobriety and spatiality in categories of lattice-valued algebras

A categorical semantics for fuzzy predicate logic

Relationship of Algebraic Theories to Powerset Theories and Fuzzy Topological Theories for Lattice-Valued Mathematics

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