Super and Hyper Products of Super Relations

Abstract

If R is a relation on X to Y, U is a relation on P (X) to Y, and V is a relation on P (X) to P (Y), then we say that R is an ordinary relation, U is a super relation, and V is a hyper relation on X to Y. Motivated by an ingenious idea of Emilia Przemska on a unified treatment of open- and closed-like sets, we shall introduce and investigate here four reasonable notions of product relations for super relations. In particular, for any two super relations U and V on X, we define two super relations U * V and U * V, and two hyper relations U ★ V and U * V on X such that : ( U * V ) ( A ) = ( A ∪ U ( A ) ) ∩ V ( A ) , ( U * V ) ( A ) = ( A ∩ U ( A ) ) ∪ U ( A ) \begin{array}{*{20}{l}} {(U*V)(A) = (A\mathop \cup \nolimits^ U(A))\mathop \cap \nolimits^ V(A),}\\ {(U*V)(A) = (A\mathop \cap \nolimits^ U(A))\mathop \cup \nolimits^ U(A)} \end{array} and ( U ★ V ) ( A ) = { B ⊆ X : ( U * V ) ( A ) ⊆ B ⊆ ( U * V ) ( A ) } , ( U * V ) ( A ) = { B ⊆ X : ( U ∩ V ) ( A ) ⊆ B ⊆ ( U ∪ V ) ( A ) } \begin{array}{*{20}{l}} {(UV)(A) = \{ B \subseteq X:\,(U*V)(A) \subseteq B \subseteq (U*V)(A)\} ,}\\ {(U*V)(A) = \{ B \subseteq X:\,(U\mathop \cap \nolimits^ V)(A) \subseteq B \subseteq (U\mathop \cup \nolimits^ V)(A)\} } \end{array} for all A ⊆ X. By using the distributivity of the operation ∩ over ∪, we can at once see that U * V ⊆ U * V. Moreover, if U ⊆ V, then we can also see that U * V = U * V. The most simple case is when U is an interior relation on X and V is the associated closure relation defined such that V (A) = U (Ac ) c for all A ⊆ X.