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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  ( A c ) c for all  A  ⊆  X .

Super and Hyper Products of Super Relations