Bijective Fuzzy Relations a Graded Approach

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 mapping 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 ≤ → ∗ ≤ ∈

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. Union: ∗ ϕ = 1 ϕ is TRUE (R / S) R / S Hence, (4) can be read as v R A R A n1 nk [∈ \∈ “It is TRUE that IF ϕ1 and ... and ϕk THEN ψ.” (R / S) R / S Explicitly using degrees, we have that v S A S A n1 nk [∈ [∈ If ϕ1 = d1 and ... and ϕk = dk then Converse inclusions have crisp counter-examples. ψ d ... d .1 ≥ 1 ∗ ∗ k 6. Residuation: R / (S / T) = (R S) / T As an example ofa classical theorem (2), we can as- ◦ sume R,S X X and consider only two antecedents: 7. Exchange: R / (S . T) = (R / S) . T ⊂ × let ϕ1 be∼ “R is a subset of S”, which means that 2.1 Graded Theorems and Their Rxy Sxy for all x,y X; let ϕ2 be interpreted as “R is≤ reflexive”, which∈ means Rxx = 1 for all x X; Reading and let the consequent ψ be “S is reflexive”. We∈ can prove the following classical theorem: Let us observe the above definitions of relational If R is a subset of S and R is reflexive then S is properties. For example the totality of R may attain reflexive. any degree from L and whenever it attains the degree 1, i.e. Tot(R) = 1, it is total in the classical sense 1Compare with (2).

44 Bijective Fuzzy Relations - A Graded Approach

It can be shown that there is alsoa graded theorem not be satisfied with use of similarities, but for exam- (Behounekˇ et al., 2008) for this statement: ple proximities (i.e., reflexive and symmetric fuzzy relations) would be enough, such as in case of Region (R S) Refl(R) Refl(S) ⊆ ∗ ≤ Connected Calculus (Batyrshin et al., 2008) that can be read analogously to the above classical theorem: 3.1 Extensionality of Set Operations “It is TRUE that IF R S and Refl(R) THEN and Relational Compositions Refl(S).” ⊆ In this case, we have incorporated also additional hid- Let us summarize some properties of extensionality den degrees for both properties: if R is reflexive to (Dankovˇ a,´ 2010c). a degree r and the degree of subsethood is equal to Proposition 2. For arbitrary F,E X Y, the fol- ⊂ × d then S is reflexive as much as R is reflexive and lowing inequalities are valid: ∼ R is a subset of S, which is expressed in degrees as Ext 1,2 (F) Ext 1,2 (E) Ext 2 (F E), (5) r d Refl(S). ≈ ∗ ≈ ≤ ≈1,2 ∩ ∗ ≤ The main difference between classical and graded Ext (F) Ext (E) Ext 2 (F E), (6) 1,2 1,2 , theorems is now obvious: having a classical theo- ≈ ∗ ≈ ≤ ≈1 2 ∪ Ext (F) Ext (E) Ext (F E). (7) rem, we use the language of classical mathematical ≈1,2 ∧ ≈1,2 ≤ ≈1,2 u logic to read the proved theorems, while in the case Readings of the above results: of graded theorems, we may read proved inequali- It is TRUE that (5) – “IF F and E are extensional ties of the form (3) directly using the generalized lan- THEN their strong intersection is extensional.” guage proposed in the above table. Let us quote from (6) – “IF F and E are extensional THEN their strong (Dankovˇ a,´ 2007a): union is extensional.” “We write classically, but we think in grades.” (7) – “IF F and E are extensional THEN their intersection is extensional.” In the following, the extensionality of supersets as 3 EXTENSIONALITY AND ITS well as similar sets will be studied. Proposition 3. For arbitrary F,E X Y we have: PROPERTIES ⊂ × 2 ∼ (F E) [Ext 1,2 (F) Ext 1,2 (E)], (8) The extensionality of a fuzzy relation F X Y w.r.t. ⊆ ≤ ≈ → ≈ ⊂ × (F E)2 [Ext (F) Ext (E)]. (9) 2 2 1,2 1,2 1 X , 2 Y is defined as follows∼: ≈ ≤ ≈ ↔ ≈ ≈ ⊂ ≈ ⊂ Observe that (8) and (9) express the extension- ∼ ∼ 2 2 ality of a higher order for Ext 1,2 w.r.t. and , Ext 1,2 (F) df ≈ ⊆ ≈ ≈ ≡ respectively. It means, extensionality when the vari- [(x x0) (y y0) Fxy Fx0y0]. ables are fuzzy relations instead of ordinary elements ≈1 ∗ ≈2 ∗ → x,x^0,y,y0 of X,Y and both (8) and (9) are valid for arbitrary F,E X Y. The extensionality is one of the most important prop- ⊂ × erties of fuzzy mathematics. It is used in fuzzy con- Readings∼ of the results: trol, fuzzy logic, fuzzy relational calculus etc., see (8) – “IF F is a subset of E (we need this requirement (Hajek,´ 1998; Belohlˇ avek,´ 2002; Perfilieva, 2001; twice) and F is extensional THEN E is extensional.” Dankovˇ a,´ 2007b). (9) – “IF F and E are similar sets (we need this re- This notion generalizes the classical one that is of quirement twice) THEN F is extensional IFF E is ex- the following form: we say that F is extensional w.r.t. tensional.” 1, 2 if The inequalities (8) and (9) together with proper- ≈ ≈ ties of relational compositions produces a long list of (x x0) (y y0) Fxy Fx0y0, ≈1 ∗ ≈2 ∗ ≤ consequences. for all x,x0 X,y,y0 Y. ∈ ∈ Corollary 4. Let 2 Let us recall that , represent generalized uni- C1 df (E1 E2) , versal and existential quantifiers, respectively; there- ≡ ⊆ V W C (F F )2, fore, the generalization is natural. Moreover, let us 2 ≡df 1 ⊆ 2 2 emphasize that we do not place any requirements on C3 df (E1 E2) , ≡ ≈ 1, 2. We would like to keep requirements as week 2 C4 df (F1 F2) . as≈ possible.≈ The reason is that some applications may ≡ ≈

45 FCTA 2016 - 8th International Conference on Fuzzy Computation Theory and Applications

Then the following inequalities are valid for arbi- Takinga closer look at the latter expression, we trary F,F ,F X Y and E,E ,E Y Z: uncover the hidden crisp equality related to the vari- 1 2 ⊂ × 1 2 ⊂ × ∼ ∼ able x. Our setting determines distinguishability on C1 [Ext (F E2) Ext (F E1)], ≤ ≈1,2 ◦ → ≈1,2 ◦ the input space by 1. Therefore we have incorporated this fact by modifying≈ the left side of the above C1 [Ext 1,2 (F / E1) Ext 1,2 (F / E2)], ≤ ≈ → ≈ inequality. Finally, we replace the universal quantifier C1 [Ext (F . E2) Ext (F . E1)], ≤ ≈1,2 → ≈1,2 with its graded generalization (as infimum) and obtain our definition. C2 [Ext 1,2 (F2 / E) Ext 1,2 (F1 / E)], ≤ ≈ → ≈ Example 5. Let L be an arbitrary MTL-algebra, C2 [Ext (F1 . E) Ext (F2 . E)], ≤ ≈1,2 → ≈1,2 f : X Y be an arbitrary function and d L. The following→ relation is functional to the degree∈ 1 w.r.t. C3 [Ext 1,2 (F E2) Ext 1,2 (F E1)], =,= ≤ ≈ ◦ ↔ ≈ ◦ the crisp equality : C3 [Ext (F / E2) Ext (F / E1)], ≤ ≈1,2 ↔ ≈1,2 d fory = f (x), C3 [Ext (F . E1) Ext (F . E2)], F1(x,y) = ≤ ≈1,2 ↔ ≈1,2 (0 otherwise.

C4 [Ext (F2 / E) Ext (F1 / E)], ≤ ≈1,2 ↔ ≈1,2 Example 6. Let X = Y = [0,1] and L be the standard C4 [Ext (F1 . E) Ext (F2 . E)]. MV-algebra, i.e., L = [0,1], ≤ ≈1,2 ↔ ≈1,2 Intersection: x y = 0 (x + y 1) ∗ df ∨ − Ext F E Ext (F E) , x y =df 1 (1 x + y). ≈1,2 ◦ ≤ ≈1,2 ◦ → ∧ − F A F A \∈ \∈ Let be a similarity on L deﬁned as: Ext (F / E) Ext F / E , ∼ ≈1,2 ≤ ≈1,2 F A F A x y = 0 (1 3 x y ). [∈ \∈ ∼ df ∨ − ·| − | Ext (F / E) Ext F / E . ≈1,2 ≤ ≈1,2 Then we ﬁnd out that: E[A E[A ∈ ∈ 1.F (x,y) =( y sin(x)) is functional to the degree 1 Union: 2 w.r.t. , . ∼ ∼ ∼ Ext 1,2 F E = Ext 1,2 (F E) , 2.F (x,y) =( y f (x)), where ≈ ◦ ≈ ◦ 3 ∼ F[A F[A ∈ ∈ 1.7x2, x [0,0.5], Ext (F / E) = Ext F / E , f (x) = ∈ ≈1,2 ≈1,2 cos(0.9x) 0.5, otherwise, F A F A − \∈ [∈ Ext (F / E) = Ext F / E . is functional to the degree 1 w.r.t. =, . ≈1,2 ≈1,2 ∼ E A E A 3.F 4(x,y) = F2 F3 is functional to the degree 0 \∈ \∈ ∨ w.r.t. , . Take x = x0 = 1 and y = 0.1, y0 = 0.8; then F∼(x∼,y) F(x ,y ) = 1, but y y = 0. ⊗ 0 0 ∼ 0 4 FUNCTIONALITY AND ITS PROPERTIES 4.1 Functionality of Relational Operations and Compositions In this section, we will introduce a property of fuzzy relation called functionality that is a direct generaliza- Let us summarize some properties of functionality tion of the related crisp notion. (Dankovˇ a,´ 2010c). Let the functionality property be given as follows: Proposition 7. The following inequalities are valid Func (F) for arbitrary F,E X Y: 1,2 df ⊂ × ≈ ≡ ∼ [(x 1 x0) Fxy Fx0y0 (y 2 y0)]. Func (F) Func (E) Func (F E), ≈ ∗ ∗ → ≈ ≈1,2 ∗ ≈2,3 ≤ ≈1,3 ◦ x,x^0,y,y0 Func 1,2 (F) Func 1,2 (E) Func 2 (F E), It generalizes the following classical notion of func- ≈ ∗ ≈ ≤ ≈1,2 ∩ tionality of F (Demirci, 1999): we say that F is func- Func 1,2 (F) Func 1,2 (E) Func 1,2 (F E). ≈ ∧ ≈ ≤ ≈ u tional w.r.t. if ≈2 We can also prove the following properties of re- F(x,y) F(x,y0) y y0, for all x X, y,y0 Y. lations analogous to those of the classical notions. ∗ ≤ ≈2 ∈ ∈

46 Bijective Fuzzy Relations - A Graded Approach

Proposition 8. It can be proved that Let us first explore how the relation between compatibility and functionality/extensionality looks like (E F)2 [Func (F) Func (E)], 1,2 1,2 for a specially chosen relation F (x,y) = y f (x). ⊆ ≤ ≈ → ≈ f df ≈2 (F E)2 [Func (F) Func (E)], ≈ ≤ ≈1,2 ↔ ≈1,2 Lemma 9. Let are valid for arbitrary F,E X Y. F (x,y) = y f (x), ⊂ × f df ≈2 ∼ 2 We can generate a long list of corollaries the rela- C df Sym( 2) (Trans( 2)) . tional compositions analogous to Corollary 4. Since ≡ ≈ ∗ ≈ the list of corollaries is identical to that of Corollary 4 Then: (only replacing Ext with Func), we do not feel the Tot( f ) Tot(F ), need to repeat it. ≤ f C [Comp , ( f ) Func 1, 2 (Ff )], ≤ ≈1 ≈2 → ≈ ≈ C [Comp , ( f ) Ext 1, 2 (Ff )]. ≤ ≈1 ≈2 → ≈ ≈ 5 FUZZY FUNCTIONS Theorem 10. Let Ff and C be as in Lemma 9, and moreover let Let us start with the graded notion of partial fuzzy function: Definition(F, f ) df [F(x, f (x)) F(x,y)], ≡ ↔ ^x _y Function 1,2 (F) df Ext 1,2 (F) Func 1,2 (F), ≈ ≡ ≈ ∧ ≈ C0 Refl( ) Definition(Ff , fF ) Tot( f ). ≡df ≈2 ∗ f ∗ which joins extensionality and functionality using . Then the following estimations are valid: Hence, the degree of beinga partial fuzzy function is∧ computed as the minimum of the degrees of these two C Comp , ( f ) Function 1, 2 (Ff ), (10) properties. Extensionality means we can substitute ≤ ≈1 ≈2 → ≈ ≈ C0 f f . (11) the original inputs by indistinguishable ones. The ≤ Ff ≈ functionality tells us that the images of indistinguish- Reading of the results: “It is TRUE that able elements are indistinguishable. Therefore, we (10) – IF is symmetric and transitive (we need ≈2 must still track what is indistinguishable, i.e., how the transitivity twice) and f is compatible, THEN we choose 1,2. These relations represent the gran- y 2 f (x) is fuzzy function.” ularity of X≈and Y, respectively, which means that ≈ (11) – IF 2 is reflexive and f is total and fFf is such they should be the coarsest relations in our system en- ≈ that for an arbitrary x: [ fF (x) 2 f (x) IFF there ex- abling the distinguishability of elements. f ≈ ists y: y 2 f (x)], THEN fFf is similar to f .” Let us fix for this section that F,Ff ,Ff X Y, ≈ F ⊂ × The reverse problem can be formulated as follows: X2, Y 2 and f , f , f : X Y. ∼ ≈1 ⊂ ≈2 ⊂ F Ff 7→ consider a fuzzy relation F and let us find a crisp func- ∼ ∼ tion fF such that it is compatible with ( 1, 2) and its 5.1 Fuzzy Functions and Their Crisp ≈ ≈ extension to fuzzy relation FfF is similar to F. Counterparts Theorem 11. Let

In classical mathematics, crisp functions can be ex- D df Tot(F) Definition(F, fF ), ≡ ∗ pressed as functional relations and vice-versa. This D0 df D Refl( 1) Sym( 2). relation is given by y = f (x) for f : X X. In fuzzy ≡ ∗ ≈ ∗ ≈ mathematics we deal with , that7→ do not need to Then it can be proved that ≈1 ≈2 be similarity or equality relations and moreover, we 2 D Func 1,2 (F) Comp ( fF ), (12) have to consider the graded compatibility property ≤ ≈ → ≈1,2 w.r.t. 1,2 (generalization of (Belohlˇ avek,´ 2002)) de- D0 Function (F) (Ff F). (13) ≤ ≈1,2 → F ≈ fined≈ as Reading of the results: “It is TRUE that ( ) ( ) ( ( ) ( )). (12) – IF F is functional and total (we need total- Comp 1, 2 f df x 1 y f x 2 f y ≈ ≈ ≡ ≈ → ≈ ity twice) and fF is such that for an arbitrary x: ^x,y [F(x, fF (x) IFF there exists y: F(x,y)] (also needed For any function f , the condition Comp=,=( f ) twice) THEN fF is the compatible function.” is trivially valid (to the degree 1). Similarly, by (13) – IF F is a total fuzzy function and fF is as above definition, Comp ( f ) = ( f (x) 2 f (x)); thus, =, 2 x and 1 is reflexive and 2 is symmetric THEN FfF is provided that is≈ reflexive, Comp ≈ ( f ) = 1 for ≈ ≈ 2 V =, 2 indistinguishable from F.” each f . ≈ ≈

47 FCTA 2016 - 8th International Conference on Fuzzy Computation Theory and Applications

T 2 T 6 FUZZY BIJECTIONS Refl( 2) (Sur(R )) Function , (R ) ≈ ∗ ∗ ≈1 2 ≤ [(R RT) ] (15) Traditionally, bijectivity refers to a mapping while ◦ ≈ ≈1 here we are dealing with fuzzy relations generally. 2 Let us look how the classical notion can be general- Refl( 1) Refl( 2) (Tot(R)) ≈ ∗ ≈ ∗ ∗ ize to fuzzy relations and whether we get analogical Function (R) ≈1,2 ≤ results. 1,2 2 2 (Bij (R) Let a,b N+, R X Y and 1 X , 2 Y ≈1,2 → ∈ ⊂ × ≈ ⊂ ≈ ⊂ T T for this section. ∼ ∼ ∼ [[(R R ) 1] [(R R) 2]]) (16) We define (a,b)-bijection as a property of a fuzzy ◦ ≈ ≈ ∧ ◦ ≈ ≈ The proof of this theorem is too long to fit the page relation R w.r.t. as follows: ≈1,2 number limitation, therefore it is omitted and left for a,b a b a full journal paper that is already in preparation. Bij (R) df (Inj (R)) (Sur(R)) . ≈1,2 ≡ 1,2 ∗ ≈ Reading of the results: “It is TRUE that The coefficients a,b refer to the numbers of times the (14) – IF 1 is reflexive and R is a surjective (needed respective properties are used in the above definition. ≈ T twice) fuzzy function THEN (R R) 2.” E.g., (2,1)-bijection is defined as: T ◦ ≈ ≈ (15) – IF 2 is reflexive and R is a surjective (needed ≈ T Inj (R) Inj (R) Sur(R). twice) fuzzy function THEN (R R ) 1.” 1,2 1,2 ◦ ≈ ≈ ≈ ∗ ≈ ∗ (16) – IF is reflexive and is reflexive and R ≈1 ≈2 It means that a relation R that is injective (to a is a total (needed twice) fuzzy function THEN IF R degree m) and surjective (to a degree n) is (2,1)- is (1,2)-bijective (because we need surjectivity twice bijective (to the degree m m n). and injectivity only once) THEN (RT R) AND ∗ ∗ 2 Let us summarize the well known results on the (R RT) .” ◦ ≈ ≈ ◦ ≈ ≈1 particular properties important for fuzzy bijections. Interesting is the fact that we have to require from Proposition 12. Let Prop be one of the properties 1 and 2 only reflexivity to prove the above inequal- ≈ ≈ Tot,Sur,Func,Inj ; then the following expressions ities. And of course, that there is a need for hav- are{ valid: } ing powers of surjectivity and totality. It is obviously T a non-trivial generalization of the following classical Tot(R) = Sur(R ) theorem (Demirci, 2000). Func (R) = Inj (RT) 1,2 1,2 Theorem 15. Let , be similarities, i.e., reflex- ≈ ≈ ≈1 ≈2 (R1 R2) Tot(R1) Tot(R2) ive, symmetric and transitive fuzzy relations. ≈ ∗ ≤ If R is a bijective total fuzzy function w.r.t. 1,2 (R1 R2) Sur(R1) Sur(R2) T T ≈ ≈ ∗ ≤ then (R R) 2 and (R R ) 1. 2 ◦ ≈ ≈ ◦ ≈ ≈ (R1 R2) Func (R1) Func (R2) ≈ ∗ ≈1,2 ≤ ≈1,2 Example 16. Let X,Y be non-empty sets of elements 2 (R R ) Inj (R ) Inj (R ) with 1 X, 2 Y, F be the set of all functions 1 2 1,2 1 1,2 2 ≈ ⊂ ≈ ⊂ ≈ ∗ ≈ ≤ ≈ ∼ ∼ Prop(R) Prop(S) Prop(R S) f : X Y and R be the set of all fuzzy relations ∗ ≤ ◦ F,F 7→X Y. Moreover, let F and R 0 ⊂ × ∼1 ⊂ ∼2 ⊂ for an arbitrary R,R1,R2 X Y,S Y Z. ∼ ∼ ∼ ⊂ × ⊂ × defined as: Guidelines for the proof∼ of this proposition∼ can be f 1 f 0 df ( f (x) 2 f 0(x)), found in (Belohlˇ avek,´ 2002). ∼ ≡ x X ≈ From the above proposition we easily deduce the ^∈ F F0 (F(x,y) F0(x,y)). following identity. ∼2 ≡df ↔ x X^,y Y Theorem 13. ∈ ∈ Define a mapping M : F R by the following as- b a a,b T 7→ (Tot(R)) (Func , (R)) = Bij (R ) signment: ∗ ≈1 2 ≈1,2 The next theorem shows relationships between M[ f ,F](x,y) = F(x,y) =( y f (x)). df ≈2 compositions of a fuzzy bijection R and RT and the In other words,M is a crisp function from F to R indistinguishability relations and . ≈1 ≈2 assigning a fuzzy relation F to a crisp function f . Theorem 14. It can be proved that M is injective provided that 2 2 is symmetric and transitive. Generally, we can Refl( 1) (Sur(R)) Function (R) ≈ ≈ ∗ ∗ ≈1,2 ≤ show that [(RT R) ] (14) 2 Sym( 2) Trans( 2) Inj (M), ◦ ≈ ≈ ≈ ∗ ≈ ≤ ∼1,2

48 Bijective Fuzzy Relations - A Graded Approach

which means that as much 2 is symmetric and tran- over a simple system of fuzzy partial propositional sitive, at least that much R≈ is injective. logic, i.e., a fuzzy propositional logic which admits To show surjectivity we incorporate (13): undeﬁned truth degrees (Behounekˇ and Novak,´ 2015). It is a topic for future research to ﬁnd some interest- D Tot(F) Function (F) (Ff 2 F), ∗ ∗ ≈1,2 ≤ F ∼ ing examples that actively work with degrees and use the presented graded theorems. where

D df Refl( 1) Sym( 2) Definition(F, fF ). ≡ ≈ ∗ ≈ ∗ ACKNOWLEDGEMENTS We would like to have = instead of 2 and that will work only with crisp properties on the∼ left of the in- The work was supported by grant No. 16–191705 equality. Thus M is surjective (to degree ) provided 1 “Fuzzy partial logic” of GA CRˇ and project LQ1602 that F is a total fuzzy function, is reflexive, is ≈1 ≈2 “IT4I XS” of MSMTˇ CR.ˇ symmetric and fF is such that y = fF (x) if and only if F(x, fF (x)) = y Y F(x,y). All these properties∈ led us to the following conclu- W sion: Let 1 be reflexive and 2 be symmetric. Then REFERENCES M is (1,1≈)-bijection to the degree≈ that is bounded from below by degrees of symmetry (a) and transitiv- Batyrshin, I., Sudkamp, T., Schockaert, S., Cock, M. D., Cornelis, C., and Kerre, E. E. (2008). Special sec- ity (b) of 2. Hence M is a (1,1)-bijection at least to degree a ≈b between F and a subset of R of total fuzzy tion: Perception based data mining and decision sup- ∗ port systems fuzzy region connection calculus: An in- functions. terpretation based on closeness. International Journal of Approximate Reasoning, 48(1):332 – 347. Behounek,ˇ L., Bodenhofer, U., and Cintula, P. (2008). Rela- 7 CONCLUSIONS tions in Fuzzy Class Theory: Initial steps. Fuzzy Sets and Systems, 159(14):1729–1772. Behounek,ˇ L. and Cintula, P. (2005). Fuzzy class theory. In this contribution, we have reviewed the results on Fuzzy Sets and Systems, 154(1):34–55. extensionality and functionality properties in the al- Behounek,ˇ L. and Novak,´ V. (2015). Towards fuzzy partial gebraic framework. Also the well known representa- logic. In Proceedings of the IEEE 45th International tion theorem for fuzzy functions has been presented Symposium on Multiple-Valued Logics (ISMVL 2015), in the graded form. Moreover, we have introduced pages 139–144. IEEE. the graded notion of a bijective relation and showed Belohlˇ avek,´ R. (2002). Fuzzy Relational Systems: Founda- graded theorems that generalize the known results of tions and Principles, volume 20 of IFSR International fuzzy mathematics. It appeared that we can relax a Series on Systems Science and Engineering. Kluwer lot of requirements on relations describing indistin- Academic/Plenum Press, New York. guishability of elements of the respective universes. Dankovˇ a,´ M. (2007a). On approximate reasoning with The main advantage of the presented approach is that graded rules. Fuzzy Sets and Systems, 158(6):652 it incorporates the whole scale for degrees of truth. – 673. The Logic of Soft ComputingThe Logic of Soft Computing IV and Fourth Workshop of the An evaluation of the degrees of the antecedents of a ERCIM working{ group} on soft computing. { } graded theorem provides the additional information Dankovˇ a,´ M. (2007b). On approximate reasoning with about the estimation of the degree of the consequent. graded rules. Fuzzy Sets and Systems, 158:652–673. We have shown that graded theorems bring a new Dankovˇ a,´ M. (2010a). Approximation of extensional fuzzy light into the already well established classical theory relations over a residuated lattice. Fuzzy Sets and Sys- of fuzzy functions and fuzzy bijective relations and tems, 161(14):1973 – 1991. Theme: Fuzzy and Un- generally, to fuzzy mathematics. An intended class of certainty Logics. applications of the introduced theory of fuzzy func- Dankovˇ a,´ M. (2010b). Representation theorem for fuzzy tions and fuzzy bijective relations is connected with functions – graded form. In Abstracts of Interna- fuzzy rules, especially, the implicative model of fuzzy tional Conference on Fuzzy Computation, pages 56– rules (Dankovˇ a,´ 2011; Stˇ epniˇ ckaˇ M., 2013). It is due 64, SciTePress - Science and Technology Publica- tions. to the fact that this model is closely connected with Dankovˇ a,´ M. (2010c). Representation theorem for fuzzy fuzzy functions; and by Theorem 14, fuzzy functions functions – graded form. In INTERNATIONAL CON- are closely related to bijections. Moreover, this re- FERENCE ON FUZZY COMPUTATION 2010, pages search is a basis for developing a theory of partial 56 – 64, Portugal. SciTePress - Science and Technol- fuzzy functions and partial bijective fuzzy relations ogy Publications.

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