Algebras from a Quasitopos of Rough Sets Anuj Kumar More Mohua Banerjee Department of Mathematics and Statistics Indian Institute of Technology Kanpur TACL Prague June 28, 2017 Algebras from a Quasitopos of Rough Sets
Outline Rough sets Categories of rough sets Generalization of categories of rough sets Algebra over subobjects of a rough set c.c.-pseudo-Boolean algebras Algebras from a Quasitopos of Rough Sets
Rough Sets (Pawlak, Z. (1982)) Rough set theory was first proposed to deal with incomplete information systems and vagueness. ( U , R ) , with U a set and R an equivalence relation over U , is called a Pawlak approximation space. For a subset X ⊆ U , consider X R := { x | [ x ] R ∩ X � = ∅} , and X R := { x | [ x ] R ⊆ X } where [ x ] R is an equivalence class in U containing x . X R is called R -upper approximation of X and X R is called R -lower approximation of X . Algebras from a Quasitopos of Rough Sets
Rough sets The pair ( X R , X R ) is called a rough set in the approximation space ( U , R ) . Here, X R ⊆ X R . Let X R and X R denote the collections of equivalence classes of R contained in X R and X R respectively, that is, � � X R = X R and X R = X R . Algebras from a Quasitopos of Rough Sets
ROUGH Category (Banerjee, M. and Chakraborty, M.K. (1993)) Objects of ROUGH have the form ( U , R , X ) , where U is a set, R an equivalence relation on U and X a subset of U . An arrow in ROUGH with dom ( U , R , X ) and cod ( V , S , Y ) is a map f : X R → Y S such that f ( X R ) ⊆ Y S . Note that the lower approximation is preserved by the arrow f . ROUGH is not a topos. Algebras from a Quasitopos of Rough Sets
ξ - ROUGH Category (Banerjee, M. and Chakraborty, M.K. (1993)) Objects of ξ - ROUGH are same as of ROUGH . A ξ - ROUGH arrow f is a ROUGH arrow with dom ( U , R , X ) and cod ( V , S , Y ) such that f ( X R / X R ) ⊆ Y S / Y S . Note that the lower approximation X R , as well as the boundary region X R / X R , is preserved by the arrow f . Algebras from a Quasitopos of Rough Sets
RSC Category (Li, X.S., Yuan, X.H. (2008)) Objects of RSC are ( X 1 , X 2 ) where X 1 , X 2 are sets and X 1 ⊆ X 2 . An RSC arrow with dom ( X 1 , X 2 ) and cod ( Y 1 , Y 2 ) is a map f : X 2 → Y 2 such that f ( X 1 ) ⊆ Y 1 . Algebras from a Quasitopos of Rough Sets
Comparison ROUGH ξ - ROUGH RSC (Li, Yuan (2008)) Objects ( U , R , X ) ( U , R , X ) ( X 1 , X 2 ) Morphisms f : X R → Y S f : X R → Y S f : X 2 → Y 2 f ( X R ) ⊆ Y S f ( X R ) ⊆ Y S f ( X 1 ) ⊆ Y 1 f ( X R \ X R ) ⊆ Y S \ Y S Theorem (More, A. K. and Banerjee, M.) ROUGH is equivalent to RSC, and forms a quasitopos. 1 ξ -ROUGH is equivalent to SET 2 , and forms a topos. 2 Algebras from a Quasitopos of Rough Sets
Subobjects and strong subobjects in RSC In figure (a), ( X 1 , X 2 ) is a subobject of ( Y 1 , Y 2 ) . In figure (b), ( X 1 , X 2 ) is a strong subobject of ( Y 1 , Y 2 ) . Hereafter, the strong subobejcts of RSC are referred as the subobjects of RSC . Algebras from a Quasitopos of Rough Sets
Generalization of RSC (More, A. K. and Banerjee, M. (2016)) The category RSC is based on sets. Let us replace sets by an arbitrary topos C . RSC ( C ) category: Objects are pairs ( A , B ) where A and B are objects in C such that there exist a monic arrow m : A → B in C . An arrow with dom ( X 1 , X 2 ) and cod ( Y 1 , Y 2 ) is a pair of arrows ( f ′ , f ) in C , such that the following diagram commutes in C . f ′ X 1 Y 1 m m ′ X 2 Y 2 f where m and m ′ are monic arrows corresponding to the objects ( X 1 , X 2 ) and ( Y 1 , Y 2 ) in RSC ( C ) . Algebras from a Quasitopos of Rough Sets
RSC ( C ) Theorem RSC ( C ) is a quasitopos. RSC ( SET ) is the category RSC . On generalizing RSC to RSC ( C ) , we have lost the Boolean property a ∨ ¬ a = 1 in the algebra of subobjects of the quasitopos RSC . Algebras from a Quasitopos of Rough Sets
An example of RSC ( C ) Consider C to be topos M-Set , where M is a monoid. An object of M-Set is a monoid action on a set X , and an arrow is a function preserving monoid action. M-Set is not a Boolean topos, when M is not a group. Consider M to be 2 = { 0 , 1 } with 0 ≤ 1. What are the objects and arrows of RSC ( 2-Set ) ? Algebras from a Quasitopos of Rough Sets
RSC ( 2-Set ) An object is a triple ( X 1 , X 2 , µ ) such that X 1 ⊆ X 2 and µ : X 2 → Y 2 is a set function such that µ 2 = µ and µ | X 1 : X 1 → X 1 . An arrow f : ( X 1 , X 2 , µ ) → ( Y 1 , Y 2 , λ ) is the set function f : X 2 → Y 2 such that f ( X 1 ) ⊆ Y 1 and λ f = f µ . f X 2 Y 2 µ λ X 2 Y 2 f RSC ( 2-Set ) gives the motivation of defining monoid action on rough sets . Algebras from a Quasitopos of Rough Sets
Monoid action on rough sets (More, A. K. and Banerjee, M. (2016)) A monoid M = ( M , ∗ , e ) action on a set X is a function λ : M × X → X satisfying λ ( e , x ) = x and λ ( m , λ ( p , x )) = λ ( m ∗ p , x ) . Definition (Monoid Action on rough sets) A monoid M action on a rough set ( X 1 , X 2 ) is a triple ( X 1 , X 2 , µ ) such that µ : M × X 2 → X 2 is a monoid action of M on the set X 2 , with the condition that µ | X 1 is a monoid action of M on X 1 . Algebras from a Quasitopos of Rough Sets
Algebra of subobjects of an RSC object Any topos (quasitopos) has an intuitionistic logic associated with the (strong) subobjects of its objects. Let M be the collection of subobjects of an RSC -object ( U 1 , U 2 ) , that is, M = { ( A 1 , A 2 ) | A 1 ⊆ U 1 , A 2 ⊆ U 2 , A 1 = U 1 ∩ A 2 } . Propositional Connectives are obtained as following: ∩ : ( A 1 , A 2 ) ∩ ( B 1 , B 2 ) = ( A 1 ∩ B 1 , A 2 ∩ B 2 ) ∪ : ( A 1 , A 2 ) ∪ ( B 1 , B 2 ) = ( A 1 ∪ B 1 , A 2 ∪ B 2 ) ¬ : ¬ ( A 1 , A 2 ) = ( U 1 \ A 1 , U 2 \ A 2 ) → : ( A 1 , A 2 ) → ( B 1 , B 2 ) = ( U 1 \ A 1 , U 2 \ A 2 ) ∪ ( B 1 , B 2 ) Algebras from a Quasitopos of Rough Sets
Observations The algebraic structure of subobjects of an object in quasitopos ROUGH is same as that of topos ξ - ROUGH . The algebra obtained is Boolean, and thus the corresponding logic obtained is classical. On a close look at negation ¬ , we see that negation is with respect to fixed RSC -object ( U 1 , U 2 ) . Therefore, we need to use the notion of relative negation in rough sets. Algebras from a Quasitopos of Rough Sets
Complementation in Rough Sets Relative rough complement, defined by Iwi´ nski (1987), in Rough sets is given by ( A 1 , A 2 ) − ( B 1 , B 2 ) = ( A 1 \ B 2 , A 2 \ B 1 ) In the lines of this, we define the negation as ∼ : ∼ ( A 1 , A 2 ) := ( U 1 \ A 2 , U 2 \ A 1 ) . This ‘results’ in a different algebraic structure on M , namely c . ∨ c . lattices. Algebras from a Quasitopos of Rough Sets
Some properties of negation ∼ ∼ 1 � = 0 ∼ ( U , U ) = ( ∅ , U \ U ) 1 ∼ 0 = 1 ∼ ( ∅ , ∅ ) = ( U , U ) 2 ∼∼ a � = a ∼∼ ( A 1 , A 2 ) = ( A 1 , A 2 ∪ ( U \ U )) 3 ∼∼∼ a = ∼ a ∼∼∼ ( A 1 , A 2 ) = ∼ ( A 1 , A 2 ) 4 a ∨ ∼ a = 1 ( A 1 , A 2 ) ∪ ∼ ( A 1 , A 2 ) = ( U , U ) 5 a ∧ ∼ a � = 0 ( A 1 , A 2 ) ∩ ∼ ( A 1 , A 2 ) = ( ∅ , A 2 \ A 1 ) 6 DeMorgan’s laws hold. 7 Algebras from a Quasitopos of Rough Sets
C . ∨ C . Lattices Definition (C.C. lattice) A contrapositionally complemented ( c . c . ) lattice is an algebra of the form ( B , ∨ , ∧ , → , ¬ , 1 ) such that the reduct ( B , ∨ , ∧ , → , 1 ) is a relatively pseudo-complemented ( r . p . c . ) lattice and ¬ satisfies the contraposition law x → ¬ y = y → ¬ x . equivalently, ¬ a = a → ¬ 1. The logic corresponding to the class of c . c . lattices is the minimal logic . Definition ( C . ∨ C . Lattices) A contrapositionally ∨ complemented ( c . ∨ c . ) lattice is a c . c . lattice satisfying x ∨ ¬ x = 1 . Algebras from a Quasitopos of Rough Sets
� � � � � � � Examples Let us consider the following 6-element r . p . c . lattice A . a ¬ a ¬ 1 a ¬ 2 a ¬ 3 a a → 0 a → w a → x a → y 1 0 1 1 1 1 y x 1 x 1 z y 1 1 y w x w 0 1 x y x y w 1 y 1 0 w x y y z Heyting c . ∨ c . c . ∨ c . 0 A does not form c . ∨ c . lattice with the negation ¬ 3 . Algebras from a Quasitopos of Rough Sets
C . ∨ C . lattices from a Boolean Algebra Consider a Boolean Algebra B = ( B , ≤ , ∨ , ∧ , ¬ , → , 0 , 1 ) and u = ( u 1 , u 2 ) where u 1 , u 2 ∈ B . Consider the set A u = { ( a 1 , a 2 ) : a 1 ≤ a 2 , a 1 , a 2 ∈ B , a 1 = a 2 ∧ u 1 } . Define the following operations on A u : ( a 1 , a 2 ) ∨ ( b 1 , b 2 ) := ( a 1 ∨ b 1 , a 2 ∨ b 2 ) ( a 1 , a 2 ) ∧ ( b 1 , b 2 ) := ( a 1 ∧ b 1 , a 2 ∧ b 2 ) ∼ ( a 1 , a 2 ) := ( u 1 ∧ ¬ a 2 , u 2 ∧ ¬ a 1 ) ( a 1 , a 2 ) → ( b 1 , b 2 ) := ( u 1 ∧ ¬ a 1 , u 2 ∧ ¬ a 2 ) ∨ ( b 1 , b 2 ) A u := ( A u , ∨ , ∧ , → , ∼ , 0 , 1 ) forms a C . ∨ C . lattice with the least element 0. Algebras from a Quasitopos of Rough Sets
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