Associatively tied implications

Abdelhamid, Abdelaziz A.; Morsi, Nehad N.

doi:10.1016/s0165-0114(02)00268-3

Cited by 21 publications

(26 citation statements)

References 17 publications

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“…See Section 3.2 below. Nevertheless, most results in [1,64] remain valid for our more general definitions, with virtually duplicate proofs.…”

Section: Adjointness Algebrasmentioning

confidence: 80%

“…We call H a comparator on L (it is called a forcing-implication in [1,64]). (iv) The three operations A, K and H are mutually related by the following condition, ∀a ∈ P , ∀y, z ∈ L:…”

Section: Adjointness Algebrasmentioning

confidence: 99%

“…Note that Definition 2.1, as well as Definition 3.1, differ from the original ones of [1,64] in that those assume one poset only (that is, P = L}. See Section 3.2 below.…”

Section: Adjointness Algebrasmentioning

confidence: 99%

“…[73,79]) were connected in order to obtain new models of multiple-valued logic, and new connectives on fuzzy statements. This research materialized in numerous articles and books, among which [1][2][3][4][6][7][8][9]12,13,[17][18][19][20][21][22][24][25][26][27][28]31,32,34,[41][42][43][44][45][46][47][49][50][51][52][53]55,56,[58][59][60][61][62][64][65][66][67][68]72,…”

Section: Adjointness Algebrasmentioning

confidence: 99%

“…We build on the work [1] of Abdel-Hamid and Morsi, in which they have enriched adjointness structures with one more conjunction, this time a binary operation T (called a tying conjunction) that ties an implication A in the following sense:…”

Section: Introductionmentioning

confidence: 99%

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The logic of tied implications, part 1: Properties, applications and representation

Morsi

Lotfallah

El-Zekey³

2006

Fuzzy Sets and Systems

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A conjunction T ties an implication operator A if the identity A(a, A(b, z)) = A(T (a, b), z) holds [We study the class of tied adjointness algebras (which are five-connective algebras on two partially ordered sets), in which the implications are tied by triangular norms. This class contains, besides residuated implications, several other implications employed in fuzzy logic. Nevertheless, we show that the algebraic inequalities of residuated algebras remain true for our tied implications, but in forms that distribute roles over the five connectives of the algebra. We apply the properties of tied implications to a generalized modus ponens inference scheme with two successive rules. We prove its equivalence to a scheme with one compound rule, when both schemata are interpreted by the compositional rule of inference, and all connectives are taken from one tied adjointness algebra. Then we quote another application of this rich theory, a notion of many-valued rough sets, which exhibit the basic mathematical behaviour of the rough sets of Pawlak. A comparator H is said to be prelinear if it satisfies H (y, z) ∨ H (z, y) = 1 for all y, z (Hájek). We introduce prelinear tied adjointness algebras, in which two comparators are prelinear. We provide a representation of those algebras, as subdirect products of tied adjointness chains, on the lines of Hájek's representation of BL-algebras. But our representations are more economical, because we employ minimal prime filters (on residuated lattices) only; rather than all prime filters.

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“…See Section 3.2 below. Nevertheless, most results in [1,64] remain valid for our more general definitions, with virtually duplicate proofs.…”

Section: Adjointness Algebrasmentioning

confidence: 80%

Section: Adjointness Algebrasmentioning

confidence: 99%

“…Note that Definition 2.1, as well as Definition 3.1, differ from the original ones of [1,64] in that those assume one poset only (that is, P = L}. See Section 3.2 below.…”

Section: Adjointness Algebrasmentioning

confidence: 99%

Section: Adjointness Algebrasmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

The logic of tied implications, part 1: Properties, applications and representation

Morsi

Lotfallah

El-Zekey³

2006

Fuzzy Sets and Systems

Self Cite

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Many-Valued Rough Sets Based on Tied Implications

El-Zekey

2014

Rough Sets and Knowledge Technology

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On Multi-adjoint Concept Lattices: Definition and Representation Theorem

Medina

Ojeda‐Aciego

Ruiz-Calviño

Formal Concept Analysis

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Abstract. Several fuzzifications of formal concept analysis have been proposed to deal with uncertainty or incomplete information. In this paper, we focus on the new paradigm of multi-adjoint concept lattices which embeds different fuzzy extensions of concept lattices, our main result being the representation theorem of this paradigm. As a consequence of this theorem, the representation theorems of the other paradigms can be proved more directly. Moreover, the multi-adjoint paradigm enriches the language providing greater flexibility to the user.

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Associatively tied implications

Cited by 21 publications

References 17 publications

The logic of tied implications, part 1: Properties, applications and representation

The logic of tied implications, part 1: Properties, applications and representation

Many-Valued Rough Sets Based on Tied Implications

On Multi-adjoint Concept Lattices: Definition and Representation Theorem

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