Notes on “pseudo-t-norms and implication operators on a complete Brouwerian lattice” and “pseudo-t-norms and implication operators: direct products and direct product decompositions”

Han, Song-Chol; Li, Hongxing

doi:10.1016/j.fss.2005.02.019

Cited by 15 publications

(13 citation statements)

References 6 publications

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“…The assumption of particular operators satisfying several properties, which imply Equation (3), was the main problem arose in [58], which was noted in [28] and clarified in [56]. Hence, the results above also fixed the impossibility of assuming these properties in particular operators.…”

Section: Corollary 27 Given the Complete Latticesmentioning

confidence: 88%

Multi-adjoint algebras versus non-commutative residuated structures

Cornejo

Medina

Ramírez‐Poussa

2015

International Journal of Approximate Reasoning

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Adjoint triples and pairs are basic operators used in several domains, since they increase the flexibility in the framework in which they are considered. This paper introduces multiadjoint algebras and several properties; also, we will show that an adjoint triple and its "dual" cannot be considered in the same framework. Moreover, a comparison among general algebraic structures used in different frameworks, which reduce the considered mathematical requirements, such as the implicative extendedorder algebras, implicative structures, the residuated algebras given by sup-preserving aggregations and the conjunctive algebras given by semi-uninorms and u-norms, is presented. This comparison shows that multi-adjoint algebras generalize these structures in domains which require residuated implications, such as in formal concept analysis, fuzzy rough sets, fuzzy relation equations and fuzzy logic.

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Section: Corollary 27 Given the Complete Latticesmentioning

confidence: 88%

Multi-adjoint algebras versus non-commutative residuated structures

Cornejo

Medina

Ramírez‐Poussa

2015

International Journal of Approximate Reasoning

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“…Consequently, the t-norm can be replaced by a weak t-norm, t-seminorm or strong pseudo-t-norm. See Fodor (1991), De Cooman andKerre (1994), Han and Li (2005), Wang and Xiong (2005) and Han et al (2006) for details. More general composite operators were also investigated in CuninghameGreen (1979CuninghameGreen ( , 1995, Kawaguchi and Miyakoshi (1998), Cheng and Peng (1988), Noskoá (2005), and Wu (2007).…”

Section: Discussionmentioning

confidence: 99%

“…associativity and commutativity, are not necessary in defining fuzzy relational equations and hence some general composite operators can be used without losing any particular property of interest. As a consequent, some generalizations have been made by Cheng and Peng (1988), Di , Kawaguchi and Miyakoshi (1998), Han and Li (2005), Li et al (2005), Wang and Xiong (2005) and Han et al (2006).…”

Section: Introductionmentioning

confidence: 98%

On the resolution and optimization of a system of fuzzy relational equations with sup-T composition

Fang

2008

Fuzzy Optim Decis Making

121

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This paper provides a thorough investigation on the resolution of a finite system of fuzzy relational equations with sup-T composition, where T is a continuous triangular norm. When such a system is consistent, although we know that the solution set can be characterized by a maximum solution and finitely many minimal solutions, it is still a challenging task to find all minimal solutions in an efficient manner. Using the representation theorem of continuous triangular norms, we show that the systems of sup-T equations can be divided into two categories depending on the involved triangular norm. When the triangular norm is Archimedean, the minimal solutions correspond one-to-one to the irredundant coverings of a set covering problem. When it is non-Archimedean, they only correspond to a subset of constrained irredundant coverings of a set covering problem. We then show that the problem of minimizing a linear objective function subject to a system of sup-T equations can be reduced into a 0-1 integer programming problem in polynomial time. This work generalizes most, if not all, known results and provides a unified framework to deal with the problem of resolution and optimization of a system of sup-T equations. Further generalizations and related issues are also included for discussion.

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“…Han et al (2006) and Abbasi discussed sup-T p equations on I = [0, 1] while Wang and Xiong (2005), and Zhang et al (2008) extended the results to the case of complete lattices, where T p is a strong pseudo-t-norm as introduced by Yu (2002, 2003) and Han and Li (2005). Note that a strong pseudo-t-norm can be either a conjunctor, a coimplicator or neither.…”

Section: As Shown In De Baetsmentioning

confidence: 93%

“…The definition of a pseudo-t-norm was proposed by Yu (2002, 2003) while Han and Li (2005) introduced the concept of a strong pseudo-t-norm by imposing a boundary condition T p (a, 0) = 0, ∀ a ∈ L, which, by Theorem 2.2, is necessary for the existence of an upper right adjoint of T p . However, a strong pseudo-t-norm may not be a conjunctor since the monotonicity in its first argument cannot be guaranteed.…”

Section: Definition 224mentioning

confidence: 99%

A survey on fuzzy relational equations, part I: classification and solvability

Fang

2009

Fuzzy Optim Decis Making

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Fuzzy relational equations play an important role in fuzzy set theory and fuzzy logic systems, from both of the theoretical and practical viewpoints. The notion of fuzzy relational equations is associated with the concept of "composition of binary relations." In this survey paper, fuzzy relational equations are studied in a general lattice-theoretic framework and classified into two basic categories according to the duality between the involved composite operations. Necessary and sufficient conditions for the solvability of fuzzy relational equations are discussed and solution sets are characterized by means of a root or crown system under some specific assumptions.

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Notes on “pseudo-t-norms and implication operators on a complete Brouwerian lattice” and “pseudo-t-norms and implication operators: direct products and direct product decompositions”

Cited by 15 publications

References 6 publications

Multi-adjoint algebras versus non-commutative residuated structures

Multi-adjoint algebras versus non-commutative residuated structures

On the resolution and optimization of a system of fuzzy relational equations with sup-T composition

A survey on fuzzy relational equations, part I: classification and solvability

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