<abstract xmlns="http://www.w3.org/1999/xhtml">

If <italic>R</italic> is a relation on <italic>X</italic> to <italic>Y, U</italic> is a relation on <italic>P</italic> (<italic>X</italic>) to <italic>Y</italic>, and <italic>V</italic> is a relation on <italic>P</italic> (<italic>X</italic>) to <italic>P</italic> (<italic>Y</italic>), then we say that <italic>R</italic> is an ordinary relation, <italic>U</italic> is a super relation, and <italic>V</italic> is a hyper relation on <italic>X</italic> to <italic>Y</italic>.
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 <italic>U</italic> and <italic>V</italic> on <italic>X</italic>, we define two super relations <italic>U</italic> * <italic>V</italic> and <italic>U</italic> * <italic>V</italic>, and two hyper relations <italic>U</italic> ★ <italic>V</italic> and <italic>U</italic> * <italic>V</italic> on <italic>X</italic> such that : <inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/tmmp-2021-0007_eq_001.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mtable columnalign="left"><mtr><mtd><mo stretchy="false">(</mo><mi>U</mi><mo>*</mo><mi>V</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mi>A</mi><mo>∪</mo><mi>U</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>∩</mo><mi>V</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>,</mo></mtd></mtr><mtr><mtd><mo stretchy="false">(</mo><mi>U</mi><mo>*</mo><mi>V</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mi>A</mi><mo>∩</mo><mi>U</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>∪</mo><mi>U</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></math><tex-math>
\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}</tex-math></alternatives></inline-formula> and <inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/tmmp-2021-0007_eq_002.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mtable columnalign="left"><mtr><mtd><mo stretchy="false">(</mo><mi>U</mi><mo>★</mo><mi>V</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>=</mo><mo>{</mo><mi>B</mi><mo>⊆</mo><mi>X</mi><mo>:</mo><mtext> </mtext><mo stretchy="false">(</mo><mi>U</mi><mo>*</mo><mi>V</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>⊆</mo><mi>B</mi><mo>⊆</mo><mo stretchy="false">(</mo><mi>U</mi><mo>*</mo><mi>V</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>}</mo><mo>,</mo></mtd></mtr><mtr><mtd><mo stretchy="false">(</mo><mi>U</mi><mo>*</mo><mi>V</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>=</mo><mo>{</mo><mi>B</mi><mo>⊆</mo><mi>X</mi><mo>:</mo><mtext> </mtext><mo stretchy="false">(</mo><mi>U</mi><mo>∩</mo><mi>V</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>⊆</mo><mi>B</mi><mo>⊆</mo><mo stretchy="false">(</mo><mi>U</mi><mo>∪</mo><mi>V</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>}</mo></mtd></mtr></mtable></math><tex-math>\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}</tex-math></alternatives></inline-formula> for all <italic>A</italic> ⊆ <italic>X</italic>.
By using the distributivity of the operation ∩ over ∪, we can at once see that <italic>U</italic> * <italic>V</italic> ⊆ <italic>U</italic> * <italic>V</italic>. Moreover, if <italic>U</italic> ⊆ <italic>V</italic>, then we can also see that <italic>U</italic> * <italic>V</italic> = <italic>U</italic> * <italic>V</italic>. The most simple case is when <italic>U</italic> is an interior relation on <italic>X</italic> and <italic>V</italic> is the associated closure relation defined such that <italic>V</italic> (<italic>A</italic>) = <italic>U</italic> (<italic>Ac</italic>)<italic>c</italic> for all <italic>A</italic> ⊆ <italic>X</italic>.
</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.

Super and Hyper Products of Super Relations

Faculty of Natural Sciences, University of Debrecen

Tatra Mountains Mathematical Publications

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

{"article-title":"Super and Hyper Products of Super Relations"}

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...

Super and Hyper Products of Super Relations

Published Online: Jan 01, 2022

Page range: 85 - 118

Received: Dec 22, 2020

DOI: https://doi.org/10.2478/tmmp-2021-0007

Keywords
relations on ordinary and power sets, unary and binary operations for relations, Galois connections, generalized uniformities, generalized open sets

© 2021 Mathematical Institute, Slovak Academy of Sciences

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

Super and Hyper Products of Super Relations

Published Online: Jan 01, 2022

Page range: 85 - 118

Received: Dec 22, 2020

DOI: https://doi.org/10.2478/tmmp-2021-0007

Keywordsrelations on ordinary and power sets, unary and binary operations for relations, Galois connections, generalized uniformities, generalized open sets

© 2021 Mathematical Institute, Slovak Academy of Sciences

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

Keywords
relations on ordinary and power sets, unary and binary operations for relations, Galois connections, generalized uniformities, generalized open sets