By Aboul Ella Hassanien, Aboul Ella Hassanien, Zbigniew Suraj, Dominik Slezak, Pawan Lingras

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Notice that a set is definable if it is a union of knowledge granules. From Proposition 4 it follows that if t is transitive, t hen Pt ( X ) = {t( x) : t( x) ⊆ X } = t∗ ( X ) a nd P t ( X ) = {t( x) : t( x) ∩ X ≠ ∅} = t∗ ( X ) . Thus operators Pt and Pt are natural generalizations of classical equivalence approximations. , both P-lower and P-upper approximations are definable sets, that is, unions of knowledge granules. Properties of operators Pt and Pt are presented in Table 8. Any set X is definable if Pt(X)=X, so X is a union of all knowledge granules contained in it.

Thus, algebra Def (Gt (U )), ∩, ∪, ∅,U is a complete distributive lattice of sets, infinitely meet distributive and infinitely join distributive. The set { A ∈ At : X ∩ A ⊆ Y } is the greatest set in Def (Gt (U )) such that and the X ∩ { A ∈ At : X ∩ A ⊆ Y } ⊆ Y set { A ∈ At : X ⊆ Y ∪ A} is the least element Z ∈ Def (Gt (U )) such that X ⊆ Y ∪ Z . From Propositions 9 and 10, we have that Def (Gt (U )), ∩, ∪, →, ∅,U is a Heyting algebra and Def (Gt (U )), ∩, ∪, ÷, ∅,U is a Brouwerian algebra.

Since X ⊆ U was chosen in an arbitrary way, Tt is a topological closure operator of the topology Def (Gt (U )). d. The family of closed sets of Alexandrov topological space (U , Def (Gt (U ))) will be denoted by Cl (Gt (U )). Note that Cl (Gt (U )) is closed for arbitrary unions and intersections (since Def (Gt (U )) is closed too). Thus Cl (Gt (U )), ∩, ∪, ∅,U is a complete distributive lattice. Theorem 10 Let (U,t) be a tolerance space and let Gt (U ) := At be a family of granules. Then the algebra of definable sets Def (Gt (U )), ∩, ∪, →, ÷, ∅,U , where operations →, ÷ are defined for any X , Y ⊆ U as follows: X → Y := { A ∈ At : X ∩ A ⊆ Y }, X ÷ Y := { A ∈ At : X ⊆ Y ∪ A}, is a complete atomic double Heyting algebra.