On involutive FL e - Algebras S. Jenei, H. Ono Univ. - - PowerPoint PPT Presentation

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On involutive FL e - Algebras S. Jenei, H. Ono Univ. - - PowerPoint PPT Presentation

Jan 15, 2024 230 likes •873 views

On involutive FL e - Algebras S. Jenei, H. Ono Univ. of Pcs, JKU JAIST 1 Terminology Commutative partially ordered monoids will be referred to as uninorms. Uninorms which are integral (resp. dually integral) will be

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On involutive FLe- Algebras

  • S. Jenei, H. Ono
  • Univ. of Pécs, JKU JAIST

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Terminology

Commutative partially ordered monoids will be referred to as uninorms. Uninorms which are integral (resp. dually integral) will be referred to as t- norms (resp. t-conorms).

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e

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e S T

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What is known about uninorms?

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e S T

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If T and S are continuous

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If T and S are continuous

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Courtesy: Fodor

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If T and S are continuous

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Courtesy: Fodor

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If T and S are continuous

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Courtesy: Fodor

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What if T and S are

  • nly residuated (left-

continuous)?

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Q1: Structural description Q2: Classification

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e S T

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e

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e S T

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e S T e S T

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e S T e S T e S T f <f <f

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e S T e S T e S T f <f <f

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Basic Definition

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Basic Definition

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Overview

Motivation beyond the algebraic interest Twin rotation construction Complete, densely ordered chains Finite Chains

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Motivation beyond the algebraic interest

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Motivation beyond the algebraic interest

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Motivation beyond the algebraic interest

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Motivation beyond the algebraic interest

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e S T

Twin rotation

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Twin rotation

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Twin rotation

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Twin rotation

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Twin rotation

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Twin rotation

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Twin rotation

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Twin rotation

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Twin rotation

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Twin rotation

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Twin rotation

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Twin rotation

THEOREM: Every conic involutive uninorm is the twin rotation of its underlying t-norm and t-conorm.

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Chains

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Complete, densely

  • rdered chains with e=f

Skew dualization on complete, densely

  • rdered posets

Classification of involutive FLe-algebras with e=f on [0,1 ]

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Complete, densely ordered chains with e=f

Skew dualization on complete, densely ordered chains

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Complete, densely ordered chains with e=f

Classification of involutive FLe-algebras with e=f on [0,1 ]

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Complete, densely ordered chains with e=f

Classification of involutive FLe-algebras with e=f on [0,1 ]

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Finite Chains

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Finite Chains

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Finite Chains

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skew dualisation between non-positive rank algebras and positive rank algebras positive rank algebras

Finite Chains

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Skew dualisation

Finite Chains

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Skew dualisation

Finite Chains

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Skew dualisation

Finite Chains

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1

  • k
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Positive rank algebras

Finite Chains

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Lemma

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Positive rank algebras

Finite Chains

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Lemma

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Positive rank algebras

Finite Chains

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Lemma

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Positive rank algebras

Finite Chains

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Lemma

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Positive rank algebras

Finite Chains

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Positive rank algebras

Finite Chains

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Positive rank algebras

Finite Chains

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Positive rank algebras

Finite Chains

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Positive rank algebras

Finite Chains

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Positive rank algebras

Finite Chains

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Positive rank algebras

Finite Chains

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Positive rank algebras

Finite Chains

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Positive rank algebras

Finite Chains

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Thank you!