A Study on Cubic <i>H</i>-Relations in a Topological Universe Viewpoint
We introduce the concrete category <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi mathvariant="bold">CRel</mi> <mi>P</mi> </msub> <mrow> <mo stretchy="false">(</mo> <...
Main Authors: | , , |
---|---|
Format: | Article |
Language: | English |
Published: |
MDPI AG
2020-04-01
|
Series: | Mathematics |
Subjects: | |
Online Access: | https://www.mdpi.com/2227-7390/8/4/482 |
_version_ | 1797571725842448384 |
---|---|
author | Jeong-Gon Lee Kul Hur Xueyou Chen |
author_facet | Jeong-Gon Lee Kul Hur Xueyou Chen |
author_sort | Jeong-Gon Lee |
collection | DOAJ |
description | We introduce the concrete category <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi mathvariant="bold">CRel</mi> <mi>P</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>H</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </semantics> </math> </inline-formula> [resp. <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi mathvariant="bold">CRel</mi> <mi>R</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>H</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </semantics> </math> </inline-formula>] of cubic <i>H</i>-relational spaces and P-preserving [resp. R-preserving] mappings between them and study it in a topological universe viewpoint. In addition, we prove that it is Cartesian closed over <inline-formula> <math display="inline"> <semantics> <mi mathvariant="bold">Set</mi> </semantics> </math> </inline-formula>. Next, we introduce the subcategory <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi mathvariant="bold">CRel</mi> <mrow> <mi>P</mi> <mo>,</mo> <mi>R</mi> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mi>H</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </semantics> </math> </inline-formula> [resp. <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi mathvariant="bold">CRel</mi> <mrow> <mi>R</mi> <mo>,</mo> <mi>R</mi> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mi>H</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </semantics> </math> </inline-formula>] of <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi mathvariant="bold">CRel</mi> <mi>P</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>H</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </semantics> </math> </inline-formula> [resp. <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi mathvariant="bold">CRel</mi> <mi>R</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>H</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </semantics> </math> </inline-formula>] and investigate it in the sense of a topological universe. In particular, we obtain exponential objects in <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi mathvariant="bold">CRel</mi> <mrow> <mi>P</mi> <mo>,</mo> <mi>R</mi> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mi>H</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </semantics> </math> </inline-formula> [resp. <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi mathvariant="bold">CRel</mi> <mrow> <mi>R</mi> <mo>,</mo> <mi>R</mi> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mi>H</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </semantics> </math> </inline-formula>] quite different from those in <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi mathvariant="bold">CRel</mi> <mi>P</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>H</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </semantics> </math> </inline-formula> [resp. <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi mathvariant="bold">CRel</mi> <mi>R</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>H</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </semantics> </math> </inline-formula>]. |
first_indexed | 2024-03-10T20:44:35Z |
format | Article |
id | doaj.art-1947d1ec56864fc7bb342851be127d98 |
institution | Directory Open Access Journal |
issn | 2227-7390 |
language | English |
last_indexed | 2024-03-10T20:44:35Z |
publishDate | 2020-04-01 |
publisher | MDPI AG |
record_format | Article |
series | Mathematics |
spelling | doaj.art-1947d1ec56864fc7bb342851be127d982023-11-19T20:23:41ZengMDPI AGMathematics2227-73902020-04-018448210.3390/math8040482A Study on Cubic <i>H</i>-Relations in a Topological Universe ViewpointJeong-Gon Lee0Kul Hur1Xueyou Chen2Division of Applied Mathematics, Nanoscale Science and Technology Institute, Wonkwang University, Iksan 54538, KoreaDepartment of Applied Mathematics, Wonkwang University, 460, Iksan-daero, Iksan-Si, Jeonbuk 54538, KoreaSchool of Mathematics, Shandong University of Technology, Zibo 255049, ChinaWe introduce the concrete category <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi mathvariant="bold">CRel</mi> <mi>P</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>H</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </semantics> </math> </inline-formula> [resp. <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi mathvariant="bold">CRel</mi> <mi>R</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>H</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </semantics> </math> </inline-formula>] of cubic <i>H</i>-relational spaces and P-preserving [resp. R-preserving] mappings between them and study it in a topological universe viewpoint. In addition, we prove that it is Cartesian closed over <inline-formula> <math display="inline"> <semantics> <mi mathvariant="bold">Set</mi> </semantics> </math> </inline-formula>. Next, we introduce the subcategory <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi mathvariant="bold">CRel</mi> <mrow> <mi>P</mi> <mo>,</mo> <mi>R</mi> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mi>H</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </semantics> </math> </inline-formula> [resp. <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi mathvariant="bold">CRel</mi> <mrow> <mi>R</mi> <mo>,</mo> <mi>R</mi> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mi>H</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </semantics> </math> </inline-formula>] of <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi mathvariant="bold">CRel</mi> <mi>P</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>H</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </semantics> </math> </inline-formula> [resp. <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi mathvariant="bold">CRel</mi> <mi>R</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>H</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </semantics> </math> </inline-formula>] and investigate it in the sense of a topological universe. In particular, we obtain exponential objects in <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi mathvariant="bold">CRel</mi> <mrow> <mi>P</mi> <mo>,</mo> <mi>R</mi> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mi>H</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </semantics> </math> </inline-formula> [resp. <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi mathvariant="bold">CRel</mi> <mrow> <mi>R</mi> <mo>,</mo> <mi>R</mi> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mi>H</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </semantics> </math> </inline-formula>] quite different from those in <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi mathvariant="bold">CRel</mi> <mi>P</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>H</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </semantics> </math> </inline-formula> [resp. <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi mathvariant="bold">CRel</mi> <mi>R</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>H</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </semantics> </math> </inline-formula>].https://www.mdpi.com/2227-7390/8/4/482cubic <i>H</i>-relational spacecubic <i>H</i>-reflexive relationtopological categorycartesian closed categorytopological universe |
spellingShingle | Jeong-Gon Lee Kul Hur Xueyou Chen A Study on Cubic <i>H</i>-Relations in a Topological Universe Viewpoint Mathematics cubic <i>H</i>-relational space cubic <i>H</i>-reflexive relation topological category cartesian closed category topological universe |
title | A Study on Cubic <i>H</i>-Relations in a Topological Universe Viewpoint |
title_full | A Study on Cubic <i>H</i>-Relations in a Topological Universe Viewpoint |
title_fullStr | A Study on Cubic <i>H</i>-Relations in a Topological Universe Viewpoint |
title_full_unstemmed | A Study on Cubic <i>H</i>-Relations in a Topological Universe Viewpoint |
title_short | A Study on Cubic <i>H</i>-Relations in a Topological Universe Viewpoint |
title_sort | study on cubic i h i relations in a topological universe viewpoint |
topic | cubic <i>H</i>-relational space cubic <i>H</i>-reflexive relation topological category cartesian closed category topological universe |
url | https://www.mdpi.com/2227-7390/8/4/482 |
work_keys_str_mv | AT jeonggonlee astudyoncubicihirelationsinatopologicaluniverseviewpoint AT kulhur astudyoncubicihirelationsinatopologicaluniverseviewpoint AT xueyouchen astudyoncubicihirelationsinatopologicaluniverseviewpoint AT jeonggonlee studyoncubicihirelationsinatopologicaluniverseviewpoint AT kulhur studyoncubicihirelationsinatopologicaluniverseviewpoint AT xueyouchen studyoncubicihirelationsinatopologicaluniverseviewpoint |