On the Structure of Semigroups on a Compact Manifold With Boundary

Mostert, Paul S.; Shields, Allen

doi:10.2307/1969668

Cited by 399 publications

(190 citation statements)

References 12 publications

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“…Table 1 shows the NM t-norm and the main continuous t-norms (Minimum, Product and Łukasiewicz) with their residua and the corresponding associated negations. The three main continuous t-norms are the basic ones since any continuous t-norm can be expressed as an ordinal sum of copies of them [51,44]. In the following proposition we summarize some basic properties of left-continuous t-norms and their residua.…”

Section: Predicate Fuzzy Logicsmentioning

confidence: 99%

On similarity in fuzzy description logics

Armengol

Dellunde

García-Cerdaña

2016

Fuzzy Sets and Systems

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This paper is a contribution to the study of similarity relations between objects represented as attribute-value pairs in Fuzzy Description Logics. For this purpose we use concrete domains in the fuzzy description logic IALCE F (D) associated either with a left-continuous or with a finite t-norm. We propose to expand this fuzzy description logic by adding a Similarity Box (SBox) including axioms expressing properties of fuzzy equalities. We also define a global similarity between objects from similarities between the values of each object attribute (local similarities) and we prove that the global similarity defined using a t-norm inherits the usual properties of the local similarities (reflexivity, symmetry or transitivity). We also prove a result relative to global similarities expressing that, in the context of the logic MTL∀, similar objects have similar properties, being these properties expressed by predicate formulas evaluated in these objects.

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Section: Predicate Fuzzy Logicsmentioning

confidence: 99%

On similarity in fuzzy description logics

Armengol

Dellunde

García-Cerdaña

2016

Fuzzy Sets and Systems

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“…The three basic continuous t-norms are the Minimum, Product and Łukasiewicz (see Table 2). These are the basic ones since any continuous t-norm can be expressed as an ordinal sum of copies of them [Mostert andShields, 1957, Ling, 1965].…”

Section: Preliminariesmentioning

confidence: 99%

A Logical Study of Local and Global Graded Similarities

Armengol

Dellunde

García-Cerdaña

2015

Applied Artificial Intelligence

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Abstract. In this paper we study the relationship between global and local similarities in the graded framework of Fuzzy Class Theory (FCT) in which already exists a graded notion of similarity. In FCT we can express the fact that a fuzzy relation is reflexive, symmetric or transitive up to a certain degree, and similarity is defined as a first-order sentence which is the fusion of three sentences corresponding to the graded notions of reflexivity, symmetry and transitivity. This allows to speak in a natural way of the degree of similarity of a relation. We consider global similarities defined from local similarities using t-norms as aggregation operators and we obtain some results in the framework of FCT that adequately interpreted allow us to say that, when we take a t-norm as aggregation operator the properties of reflexivity, symmetry and transitivity of fuzzy binary relations at the global level inherit the properties of the fuzzy binary relations at the local level when we fusion them, and that the global similarity is a congruence if some of the local similarities are congruences.

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“…We may write x M for xf if T is clear from context. Continuous t-norms turn [0, 1] into a topological semigroup [3] (more precisely, into an /-semigroup [7,26,28]). We have the following representations [18,23,31], using the pseudo-inverse f …”

Section: Introductionmentioning

confidence: 99%

“…Such left-continuous t-norms can be characterized in very special cases only, for example, if their Archimedean components satisfy the cancellation law in which case they are generated because of [1]. However, if the Archimedean components of such left-continuous t-norms cannot be extended to /-semigroups [7,26,28] or if they do not satisfy the cancellation law, the characterization of those t-norms is still an open problem.…”

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confidence: 99%

Archimedean components of triangular norms

Klement¹,

Mesiar²,

Pap³

2005

J. Aust. Math. Soc.

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The Archimedean components of triangular norms (which turn the closed unit interval into an abelian, totally ordered semigroup with neutral element 1) are studied, in particular their extension to triangular norms, and some construction methods for Archimedean components are given. The triangular norms which are uniquely determined by their Archimedean components are characterized. Using ordinal sums and additive generators, new types of left-continuous triangular norms are constructed.2000 Mathematics subject classification: primary 20M05; secondary 03E72.

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On the Structure of Semigroups on a Compact Manifold With Boundary

Cited by 399 publications

References 12 publications

On similarity in fuzzy description logics

On similarity in fuzzy description logics

A Logical Study of Local and Global Graded Similarities

Archimedean components of triangular norms

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