Arrow Categories Michael Winter Department of Computer Science - PowerPoint PPT Presentation

Arrow Categories Michael Winter Department of Computer Science Brock University St. Catharines, Canada mwinter@brocku.ca

Content 1. Binary (Boolean valued), fuzzy and L -fuzzy relations 2. Dedekind categories (Boolean valued relations) 3. Goguen categories (Fuzzy/ L -fuzzy relations) 4. Arrow categories

Binary (Boolean valued) relation (Category Rel)   1 0 0    1 0 1      0 1 1 Fuzzy relation (Category Rel([0,1]))   0 . 1 0 . 8 0 . 0   1 . 0 0 . 4 0 . 9       0 . 0 0 . 2 0 . 1

L -fuzzy relation ( L a complete distributive lattice, Category Rel( L )) 1 � ❅ � ❅ � ❅ m   l k l 1 ❅ � ❅ �   k m ❅ �  0    k   l 0 1 L = 0

Dedekind categories Definition: A Dedekind category R is a category satisfying the following: 1. For all objects A and B the collection R [ A , B ] is a complete distributive lattice (complete Heyting algebra). Meet, join, the induced ordering, the least and the greatest element are denoted by ⊓ , ⊔ , ⊑ , ⊥ ⊥ AB , ⊤ ⊤ AB , respectively. 2. There is a monotone operation � (called converse) such that for all relations Q : A → B and R : B → C the following holds � = Q . ( Q ; R ) � = R � ; Q � , ( Q � ) 3. For all relations Q : A → B , R : B → C and S : A → C the

modular law holds: Q ; R ⊓ S ⊑ Q ; ( R ⊓ Q � ; S ) . 4. For all relations R : B → C and S : A → C there is a relation S / R : A → B (called the left residual of S and R ) such that for all Q : A → B the following holds Q ; R ⊑ S Q ⊑ S / R . ⇐ ⇒

Definition (Matrix category) Let R be a Dedekind category. The category R + of matrices with coefficients from R is defined by: 1. The class of objects of R + is the collection of all functions from an arbitrary set I into the class of objects Obj R of R . 2. For every pair f : I → Obj R , g : J → Obj R of objects from R + , a morphism R : f → g is a function from I × J into the class of all morphisms Mor R of R such that R ( i , j ) : f ( i ) → g ( j ) holds. 3. For R : f → g and S : g → h composition is defined by ( R ; S )( i , k ) : = R ( i , j ) ; S ( j , k ) . � j ∈ J

4. For R : f → g conversion defined by R � ( j , i ) : = ( R ( i , j )) � . 5. For R , S : f → g join and meet are defined by ( R ⊔ S )( i , j ) R ( i , j ) ⊔ S ( i , j ) , : = ( R ⊓ S )( i , j ) R ( i , j ) ⊓ S ( i , j ) . : = 6. The identity, zero and universal elements are defined by  ⊥ f ( i 1 ) f ( i 2 ) : i 1 � = i 2 ⊥  I f ( i 1 , i 2 ) : = : i 1 = i 2 , I f ( i 1 )  ⊥ fg ( i , j ) ⊥ : = ⊥ f ( i ) g ( j ) , ⊥ ⊤ fg ( i , j ) ⊤ : = ⊤ f ( i ) g ( j ) . ⊤

Some results Lemma: R + is a Dedekind category. Corollary: Let L = ( L , ∨ , ∧ , 0 , 1 ) be a complete distributive lattice with least element 0 and greatest element 1. Then L is an one-object Dedekind category with identity 1 and composition ∧ (the residual is given by the pseudo-complement). Consequently, L + is a Dedekind category, called the full category of L -relations.

Lemma: The collection of scalar relations on A , i.e., the relations k : A → A with k ⊑ I A and ⊤ ⊤ AA ; k = k ; ⊤ ⊤ AA , constitutes a complete distributive lattice.   k 0 0   k Example:  0 0      k 0 0 Theorem: There is no formula ϕ in the language of Dedekind categories such that for all lattices L and L -relations R : A → B we have L + | = ϕ [ R ] ⇐ ⇒ R is 0-1 crisp .

Goguen categories Definition: A Goguen category G is a Dedekind category with ⊤ AB for all objects A and B together with two operations ↑ ⊥ AB � = ⊤ ⊥ and ↓ satisfying the following: 1. R ↑ , R ↓ : A → B for all R : A → B . 2. ( ↑ , ↓ ) is a Galois correspondence, i.e., R ↑ ⊑ S ⇐ ⇒ R ⊑ S ↓ for all R , S : A → B . ↑ = R ↑ � ; S ↓ for all R : B → A and S : B → C . 3. ( R � ; S ↓ ) ⊥ AA is a nonzero scalar then α ↑ = I A . 4. If α � = ⊥

1 � ❅ � ❅ � ❅ m l   k l ❅ � 1 ❅ �   ❅ � k m  0  k     l L = 0 1 0 ↑ ↓         k l k l 1 1 1 1 1 1 0 0         k m k m =  , =  0   0 1 1   0   0 0 0                 l l 0 1 0 1 1 0 1 0 1 0

α ∈ M f ( α ) for all sets of 4. For all functions f so that f ( � M ) = scalars and f ( α ) ↑ = f ( α ) for all scalars the following equivalence holds ( α \ R ) ↓ ⊑ f ( α ) for all scalars α . α ; f ( α ) R ⊑ � ⇐ ⇒ α : A → A α scalar ↓ ↓           l k l k 0 0 1 1 1 1 0 1           l k m k m  \ = =           0 0 0 0 0 0 0                    l l 0 0 0 1 0 1 1 0 1 1

Some results Theorem: Let L be a complete distributive lattice. Then L + together with the operations  1 iff R ( x , y ) � = 0  R ↑ ( x , y ) : = , 0 iff R ( x , y ) = 0   1 iff R ( x , y ) = 1  R ↓ ( x , y ) : = , 0 iff R ( x , y ) � = 1  is a Goguen category. Furthermore, for a relation R in L + we have R ↑ = R iff R 0-1 crisp.

Lemma: For each pair of objects A and B the set of scalar elements on A resp. on B are isomorphic lattices. Lemma: Let G be a Goguen category and R : A → B be a relation. Then we have α A ; ( α A \ R ) ↓ = R , � 1. α scalar ( α A \ R ) ↓ = R ↑ . � 2. α A scalar α A � = ⊥ ⊥ AA

Theorem (Pseudo-representation Theorem): Every Goguen category G is isomorphic to the category of antimorphisms mapping the scalars of G to the crisp relations of G . Corollary: A Goguen category is representable iff its subcategory of crisp relations is representable.

Further results/studies of Goguen categories 1. Definability of norm-based operations; 2. Validity of certain formulae in the subcategory of crisp relations; 3. Applications in computer science, e.g., fuzzy controller; 4. ...

Arrow categories Definition: An arrow category A is a Dedekind category with ⊥ AB for all objects A and B together with two operations ↑ ⊤ AB � = ⊥ ⊤ and ↓ satisfying the following: 1. R ↑ , R ↓ : A → B for all R : A → B . 2. ( ↑ , ↓ ) is a Galois correspondence. ↑ = R ↑ � ; S ↓ for all R : B → A and S : B → C . 3. ( R � ; S ↓ ) ↑ = Q ↑ ⊓ R ↓ for all Q , R : A → B . 4. ( Q ⊓ R ↓ ) ⊥ AA is a non-zero scalar then α ↑ 5. If α A � = ⊥ A = I A .

Lemma: For each pair of objects A and B the set of scalar elements on A resp. on B are isomorphic lattices. Lemma: Let A be an arrow category and R : A → B be a relation. Then we have α A ; ( α A \ R ) ↓ ⊑ R , � 1. α scalar ( α A \ R ) ↓ ⊑ R ↑ . � 2. α A scalar α A � = ⊥ ⊥ AA

� � 1 1 Example 1: 1 1 � � b 1 b 1 � � ������ � � � � � � � � a b L 1 1 b a 1 1 � � ������ � � � � � � a 1 1 a 1 � ������ � � � � � � � � � a a 1 0 b a a 0 1 � � ������ � � � � � � a a 0 a 0 � � 0 0 0 0 0

� � 1 1 1 1 � � b 1 Example 2: b 1 � � � � � � � � � � b b a 1 b b a 1 � � � � � � � � � � � � � b a 1 0 a b 0 1 � � � � � � � � b 0 b 0 � � 0 0 0 0

Arrow categories with cuts Definition: An arrow category with cuts A is an arrow category so that α A ; ( α A \ R ) ↓ R ⊑ � α scalar for all relations R : A → B holds.

� � � � Example x 0 � � x 1 1 1 = ⊤ ⊤ 1 1 x 2 � � 1 0 R = I 0 1 L x ∞ � � 0 0 = ⊥ ⊥ 0 0 0

Thank you for your attention.