Bijective Fuzzy Relations A Graded Approach Martina Dankovˇ a´ Institute for Research and Applications of Fuzzy Modeling, CE IT4Innovations, University of Ostrava, 30. dubna 22, Ostrava, Czech Republic Keywords: Fuzzy Relations, Fuzzy Functions, Partial Fuzzy Functions, Bijective Mappings, Fuzzy Class Theory. Abstract: Bijectivity is one of crucial mathematical notions. In this paper, we will present a fuzzy bijective mapping as a fuzzy relation that has several special properties. These properties come with degrees and so the bijectivity is also a graded property. We will focus on properties of this type of relations and show graded versions of theorems on fuzzy bijections that are known from traditional Fuzzy Set Theory. 1 INTRODUCTION to make an implicative model of fuzzy IF–THEN rules. In (Dankovˇ a,´ 2010b), the framework was In this work, we will present a gradual version of that of Fuzzy Class Theory (Behounekˇ and Cintula, results on fuzzy bijective functions from (Demirci, 2005). It enables formulation of notions in the stan- 2000; Demirci, 2001). The presented results dard mathematical notation whereas the background lay foundations for developing a theory of partial machinery provides the non-classical interpretation. fuzzy functions and partial bijective fuzzy relations Since this framework might be difficult to read for over a simple system of fuzzy partial propositional fuzzy mathematicians, we choose the common alge- logic (Behounekˇ and Novak,´ 2015). There we deal braic one for this paper and, in addition, review results with membership functions that admit undefined truth presented in (Dankovˇ a,´ 2010b). degrees. A basis for the notion of fuzzy bijective function is the notion of fuzzy function. This notion took 2 BASIC NOTIONS many forms by their definitions; e.g., it is any map- ping that assigns a fuzzy set to a fuzzy set or fuzzy set In the following, we will work with algebraic struc- to a point (Novak´ et al., 1999); it is a fuzzy relation tures used as basic structures for fuzzy logic of left- that meets two properties, namely, extensionality and continuous t-norms, the so called monoidal t-norm functionality for a partial fuzzy function, and if it is based logic. in addition total then it is called a perfect fuzzy func- Definition 1. An MTL-algebra L is a bounded resid- tion (Demirci, 1999; Demirci, 2001; Demirci and Re- uated lattice casens, 2004; Perfiljeva I., 2014); etc. Overviews to- L = L, , , , ,0,1 (1) gether with applications can be found in the following h ∨ ∧ ∗ → i exemplary sources (Klawonn, 2000; Demirci, 2001; with four binary operations and two constants such Belohlˇ avek,´ 2002). that As noted in (Demirci, 2000), not all notions of L = L, , ,0,1 is a lattice with the largest el- • h ∨ ∧ i fuzzy function do coincide with the classical notion ement 1 and the least element 0 w.r.t. the lattice for crisp functions. We will avoid this problem and ordering , ≤ all the subsequent definitions will be consistent with L = L, ,1 is a commutative semigroup with the the classical notion whenever applied on crisp inputs. • unit elementh ∗ i1, i.e., is commutative, associative, We follow up (Dankovˇ a,´ 2010a) where the graded and 1 x = x for all∗ x L, ∗ ∈ property of extensionality has been studied and and form an adjoint pair, i.e., (Dankovˇ a,´ 2010b; Dankovˇ a,´ 2011) where the graded • ∗ → z (x y) iff x z y for all x,y,z L, property of functionality has been explored and used ≤ → ∗ ≤ ∈ 42 Danková,ˇ M. Bijective Fuzzy Relations - A Graded Approach. DOI: 10.5220/0006053300420050 In Proceedings of the 8th International Joint Conference on Computational Intelligence (IJCCI 2016) - Volume 2: FCTA, pages 42-50 ISBN: 978-989-758-201-1 Copyright c 2016 by SCITEPRESS – Science and Technology Publications, Lda. All rights reserved Bijective Fuzzy Relations - A Graded Approach satisfying the pre-linearity equation: Set-similarity: • • (x y) (y x) = 1. R S (R S) (S R) → ∨ → ≈ ≡df ⊆ ∗ ⊆ In the sequel, let us assume L be an MTL-algebra Totality: of the form (1). We will call the operation product • ∗ Tot(R) Rxy and residuum. Moreover, we define the operations ≡df called→bi-residuum and powers w.r.t. the product : ^x _y ∗ Surjectivity: x y =df (x y) (y x) • ↔ n → ∧ → ϕ =df ϕ ... ϕ Sur(R) df Rxy ∗ ∗ ≡ y x n-times ^_ To reduce the number of parenthesis used in mathe- Injectivity: | {z } • matical expressions we set that has the highest pri- ∗ ority and the lowest priority out of all operations Inj (R) df ≈1,2 ≡ that are at→ the disposal. [((y y0) Rxy Rx0y0) (x x0)] Throughout the whole text, we will deal with ≈2 ∗ ∗ → ≈1 x,x ,y,y fuzzy relations whose membership functions are de- ^0 0 fined on non-empty sets and take values from the sup- In the sequel, we will freely use the class notation port of L, and will denote this fact by . In the se- ⊂ that has been formally developed in (Behounekˇ and quel, let X,Y = 0/, R,S X Y, ∼ X X and Cintula, 2005). It means that for a fuzzy set A X 6 ⊂ × ≈1 ⊂ × ⊂ 2 Y Y are fuzzy relations∼ such that∼ all infima ∼ ≈ ⊂ × x Ax respresents A and∼ suprema needed for the definition of the truth- { | } y x Ax stands for Ay value of any expression exist in L. ∈ { | } Which variables belong to which sets is always and for a fuzzy relation R X Y clear from the context. Therefore it is not necessary ⊂ × ∼ to specify membership of variables in sets in expres- xy Rxy respresents R sions, including infima and suprema; e.g., we write { | } x0y0 xy Rxy stands for Rx0y0 x,y R(x,y) instead of x X,y Y R(x,y). And for the ∈ { | } sake of brevity we use Rxy∈ instead∈ of R(x,y). W W and analogously we proceed for an arbitrary expres- Let us define the following graded properties of sion ϕ. fuzzy relations: Let us introduce the following relational opera- Reflexivity: tions: • T R =df yx Rxy inverse Refl(R) df Rxx { | } ≡ R S =df xy Rxy Sxy strong intersection ^x ∩ { | ∗ } R S =df xy Rxy Sxy union Symmetry: R t S = {xy | Rxy ∨ Sxy} intersection • u df { | ∧ } Sym(R) (Rxy Ryx) We will additionally deal with relational composi- ≡df → ^x,y tions, defined using a class notation. A systematic Transitivity: study can be find in (Belohlˇ avek,´ 2002). We will use • three basic relational compositions: Trans(R) [(Rxy Ryz) Rxz] ≡df ∗ → sup-T composition: x^,y,z • Similarity: R S =df xy (Rxz Szy) • ◦ { | z ∗ } _ Sim(R) df Refl(R) Sym(R) Trans(R) ≡ ∗ ∗ BK-subproduct: Subsethood: • • R / S =df xy (Rxz Szy) R S df (R(x,y) S(x,y)) { | → } ⊆ ≡ → ^z ^x,y Strong set-similarity: BK-superproduct: • • R . S = xy (Szy Rxz) R = S df (R(x,y) S(x,y)) df ∼ ≡ ↔ { | z → } ^x,y ^ 43 FCTA 2016 - 8th International Conference on Fuzzy Computation Theory and Applications The crisp identity = and crisp inclusion of and we have the correspondence with the classical no- fuzzy relation are defined standardly: v tion of totality: for all x X there exists y Y such that Rxy = 1. For an arbitrary∈ property ϕ introduced∈ = = , R S if Rxy Sxy, for all x y; in this paper, it is valid that if ϕ = 1 then ϕ is identical R S if Rxy Sxy, for all x,y. to the correspondent classical property. v ≤ Let the following fuzzy relations be of the appro- Statements that will be presented in this paper are priate types; then we can summarize properties of called gradual or graded theorems. It means that in- classical theorem sup-T-composition (Belohlˇ avek,´ 2002): stead of a of fuzzy mathematics If (ϕ = 1) and ... and (ϕ = 1) then (ψ = 1) (2) 1. Transposition: (R S)T = ST RT 1 k ◦ ◦ we search fora more informative and general (non- 2. Monotony: (R R ) (R S R S) 1 ⊆ 2 ≤ 1 ◦ ⊆ 2 ◦ equivalent) form of this statement, the so called 3. Union: R S = (R S) graded theorem: ◦ ◦ n nk R A R A ϕ 1 ... ϕ ψ. (3) [∈ [∈ 1 ∗ ∗ k ≤ By the properties of , it is equivalent to 4. Intersection: R S (R S) → ◦ v ◦ n1 nk R A R A (ϕ ... ϕ ψ) = 1 (4) \∈ \∈ 1 ∗ ∗ k → 5. Associativity: (R S) T = R (S T) where interprets strong conjunction and ϕ ,...,ϕ , ◦ ◦ ◦ ◦ ∗ 1 k Properties of BK-products: ψ represent the formalization of premises and the con- T T T clusion in a form that enjoys degrees of truths. Part of 1. Transposition: (R / S) = S . R the analysis is finding out how many times the an- 2. Monotony of /: tecedents ϕ1,...,ϕk need be used to provide a lower bound for the degree of the consequent ψ; the result (R1 R2) (R2 / S R1 / S) ⊆ ≤ ⊆ is encoded in the degrees n1,...,nk. (S1 S2) (R / S1 R / S2) Graded theorems seem to be difficult for non- ⊆ ≤ ⊆ experienced readers; therefore, some translations will 3. Monotony of .: be added as a guideline. The proposed reading of (R1 R2) (R1 . S R2 . S) graded theorems is analogous to the classical case (as ⊆ ≤ ⊆ when using classical mathematical logic) and it is dis- (S S ) (R . S R . S ) 1 ⊆ 2 ≤ 2 ⊆ 1 tinguished by a special typeface; for the chosen op- 4. Intersection: erations of L, we set: Expression Reading (R / S) = R / S ϕ ψ IF ϕ THEN ψ → R\A R[A ϕ ψ ϕ IFF ψ ∈ ∈ ↔ (R / S) = R / S ϕ ψ ϕ AND ψ ∧ S A S A ϕ ψ ϕ OR ψ \∈ \∈ ∨ ϕ ψ ϕ and ψ 5.