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: | , , |
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| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2020-04-01
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| Series: | Mathematics |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2227-7390/8/4/482 |
| _version_ | 1797571725842448384 |
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| 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 |
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