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

On involutive FL e - Algebras S. Jenei, H. Ono Univ. of Pécs, JKU JAIST 1

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). 2

e 3

e S T 4

What is known about uninorms? e S T 5

If T and S are continuous 6

If T and S are continuous Courtesy: Fodor 7

If T and S are continuous Courtesy: Fodor 8

If T and S are continuous Courtesy: Fodor 9

What if T and S are only residuated (left- continuous)?

Q1: Structural description Q2: Classification e S T 11

e 12

e S T 13

e e S S T T 14

e e e S S S <f f <f T T T 15

e e e S S S <f <f T T T f 16

Basic Definition 17

Basic Definition 18

Overview Motivation beyond the algebraic interest Twin rotation construction Complete, densely ordered chains Finite Chains 19

Motivation beyond the algebraic interest 20

Motivation beyond the algebraic interest 21

Motivation beyond the algebraic interest 22

Motivation beyond the algebraic interest 23

Twin rotation e S T 24

Twin rotation 25

Twin rotation 26

Twin rotation 27

Twin rotation 28

Twin rotation 29

Twin rotation 30

Twin rotation 31

Twin rotation 32

Twin rotation 33

Twin rotation 34

Twin rotation THEOREM: Every conic involutive uninorm is the twin rotation of its underlying t-norm and t-conorm. 35

Chains 36

Complete, densely ordered chains with e=f Skew dualization on complete, densely ordered posets Classification of involutive FLe-algebras with e=f on [0,1 ] 37

38

Complete, densely ordered chains with e=f Skew dualization on complete, densely ordered chains 39

Complete, densely ordered chains with e=f Classification of involutive FLe-algebras with e=f on [0,1 ] 40

Complete, densely ordered chains with e=f Classification of involutive FLe-algebras with e=f on [0,1 ] 41

Finite Chains 42

Finite Chains 43

Finite Chains 44

Finite Chains skew dualisation between non-positive rank algebras and positive rank algebras positive rank algebras 45

Finite Chains Skew dualisation 46

Finite Chains Skew dualisation 47

Finite Chains Skew dualisation 1 -k 48

Finite Chains Positive rank algebras Lemma 49

Finite Chains Positive rank algebras Lemma 50

Finite Chains Positive rank algebras Lemma 51

Finite Chains Positive rank algebras Lemma 52

Finite Chains Positive rank algebras 53

Finite Chains Positive rank algebras 54

Finite Chains Positive rank algebras 55

Finite Chains Positive rank algebras 56

Finite Chains Positive rank algebras 57

Finite Chains Positive rank algebras 58

Finite Chains Positive rank algebras 59

Finite Chains Positive rank algebras 60

Finite Chains Positive rank algebras 61

Thank you!