Isomorphism Theorems in the Primary Categories of Krasner Hypermodules
Let <i>R</i> be a Krasner hyperring. In this paper, we prove a factorization theorem in the category of Krasner <i>R</i>-hypermodules with inclusion single-valued <i>R</i>-homomorphisms as its morphisms. Then, we prove various isomorphism theorems for a smaller ca...
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MDPI AG
2019-05-01
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| Series: | Symmetry |
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| Online Access: | https://www.mdpi.com/2073-8994/11/5/687 |
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| author | Hossein Shojaei Dario Fasino |
| author_facet | Hossein Shojaei Dario Fasino |
| author_sort | Hossein Shojaei |
| collection | DOAJ |
| description | Let <i>R</i> be a Krasner hyperring. In this paper, we prove a factorization theorem in the category of Krasner <i>R</i>-hypermodules with inclusion single-valued <i>R</i>-homomorphisms as its morphisms. Then, we prove various isomorphism theorems for a smaller category, i.e., the category of Krasner <i>R</i>-hypermodules with strong single-valued <i>R</i>-homomorphisms as its morphisms. In addition, we show that the latter category is balanced. Finally, we prove that for every strong single-valued <i>R</i>-homomorphism <inline-formula> <math display="inline"> <semantics> <mrow> <mi>f</mi> <mo lspace="0pt">:</mo> <mi>A</mi> <mo>→</mo> <mi>B</mi> </mrow> </semantics> </math> </inline-formula> and <inline-formula> <math display="inline"> <semantics> <mrow> <mi>a</mi> <mo>∈</mo> <mi>A</mi> </mrow> </semantics> </math> </inline-formula>, we have <inline-formula> <math display="inline"> <semantics> <mrow> <mi>K</mi> <mi>e</mi> <mi>r</mi> <mo>(</mo> <mi>f</mi> <mo>)</mo> <mo>+</mo> <mi>a</mi> <mo>=</mo> <mi>a</mi> <mo>+</mo> <mi>K</mi> <mi>e</mi> <mi>r</mi> <mo>(</mo> <mi>f</mi> <mo>)</mo> <mo>=</mo> <mo>{</mo> <mi>x</mi> <mo>∈</mo> <mi>A</mi> <mo>∣</mo> <mi>f</mi> <mo>(</mo> <mi>x</mi> <mo>)</mo> <mo>=</mo> <mi>f</mi> <mo>(</mo> <mi>a</mi> <mo>)</mo> <mo>}</mo> </mrow> </semantics> </math> </inline-formula>. |
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| id | doaj.art-087286e7a6514358add710e9a3a7a4d3 |
| institution | Directory Open Access Journal |
| issn | 2073-8994 |
| language | English |
| last_indexed | 2024-04-11T11:12:31Z |
| publishDate | 2019-05-01 |
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| series | Symmetry |
| spelling | doaj.art-087286e7a6514358add710e9a3a7a4d32022-12-22T04:27:26ZengMDPI AGSymmetry2073-89942019-05-0111568710.3390/sym11050687sym11050687Isomorphism Theorems in the Primary Categories of Krasner HypermodulesHossein Shojaei0Dario Fasino1School of Mathematics, Statistics and Computer Science, College of Science, University of Tehran, Tehran P.O. Box 14155-6455, IranDepartment of Mathematics, Computer Science and Physics, University of Udine, 33100 Udine, ItalyLet <i>R</i> be a Krasner hyperring. In this paper, we prove a factorization theorem in the category of Krasner <i>R</i>-hypermodules with inclusion single-valued <i>R</i>-homomorphisms as its morphisms. Then, we prove various isomorphism theorems for a smaller category, i.e., the category of Krasner <i>R</i>-hypermodules with strong single-valued <i>R</i>-homomorphisms as its morphisms. In addition, we show that the latter category is balanced. Finally, we prove that for every strong single-valued <i>R</i>-homomorphism <inline-formula> <math display="inline"> <semantics> <mrow> <mi>f</mi> <mo lspace="0pt">:</mo> <mi>A</mi> <mo>→</mo> <mi>B</mi> </mrow> </semantics> </math> </inline-formula> and <inline-formula> <math display="inline"> <semantics> <mrow> <mi>a</mi> <mo>∈</mo> <mi>A</mi> </mrow> </semantics> </math> </inline-formula>, we have <inline-formula> <math display="inline"> <semantics> <mrow> <mi>K</mi> <mi>e</mi> <mi>r</mi> <mo>(</mo> <mi>f</mi> <mo>)</mo> <mo>+</mo> <mi>a</mi> <mo>=</mo> <mi>a</mi> <mo>+</mo> <mi>K</mi> <mi>e</mi> <mi>r</mi> <mo>(</mo> <mi>f</mi> <mo>)</mo> <mo>=</mo> <mo>{</mo> <mi>x</mi> <mo>∈</mo> <mi>A</mi> <mo>∣</mo> <mi>f</mi> <mo>(</mo> <mi>x</mi> <mo>)</mo> <mo>=</mo> <mi>f</mi> <mo>(</mo> <mi>a</mi> <mo>)</mo> <mo>}</mo> </mrow> </semantics> </math> </inline-formula>.https://www.mdpi.com/2073-8994/11/5/687Krasner hyperringKrasner hypermoduleisomorphism theoremfactorization theoremcategory theory |
| spellingShingle | Hossein Shojaei Dario Fasino Isomorphism Theorems in the Primary Categories of Krasner Hypermodules Symmetry Krasner hyperring Krasner hypermodule isomorphism theorem factorization theorem category theory |
| title | Isomorphism Theorems in the Primary Categories of Krasner Hypermodules |
| title_full | Isomorphism Theorems in the Primary Categories of Krasner Hypermodules |
| title_fullStr | Isomorphism Theorems in the Primary Categories of Krasner Hypermodules |
| title_full_unstemmed | Isomorphism Theorems in the Primary Categories of Krasner Hypermodules |
| title_short | Isomorphism Theorems in the Primary Categories of Krasner Hypermodules |
| title_sort | isomorphism theorems in the primary categories of krasner hypermodules |
| topic | Krasner hyperring Krasner hypermodule isomorphism theorem factorization theorem category theory |
| url | https://www.mdpi.com/2073-8994/11/5/687 |
| work_keys_str_mv | AT hosseinshojaei isomorphismtheoremsintheprimarycategoriesofkrasnerhypermodules AT dariofasino isomorphismtheoremsintheprimarycategoriesofkrasnerhypermodules |