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
)
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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.
On Functions Preserving Products of Certain Classes of Semimetric Spaces
