Coalgebraic Structure of Genetic Inheritance
Although in the broadly defined genetic algebra, multiplicationsuggests a forward direction of from parents to progeny, when looking from the reversedirection, it also suggests to us a new algebraic structure ---coalgebraic structure, which we call genetic coalgebras. It isnot the dual coalgebraic...
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AIMS Press
2004-06-01
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| Series: | Mathematical Biosciences and Engineering |
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| Online Access: | https://www.aimspress.com/article/doi/10.3934/mbe.2004.1.243 |
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| author | Jianjun Tian Bai-Lian Li |
| author_facet | Jianjun Tian Bai-Lian Li |
| author_sort | Jianjun Tian |
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| description | Although in the broadly defined genetic algebra, multiplicationsuggests a forward direction of from parents to progeny, when looking from the reversedirection, it also suggests to us a new algebraic structure ---coalgebraic structure, which we call genetic coalgebras. It isnot the dual coalgebraic structure and can be used in theconstruction of phylogenetic trees. Mathematically, to constructphylogenetic trees means we need to solve equations x[n]=a, or x(n)=b. It is generally impossible tosolve these equations in algebras. However, we can solve themin coalgebras in the sense of tracing back for their ancestors. Athorough exploration of coalgebraic structure in genetics isapparently necessary. Here, we develop a theoretical frameworkof the coalgebraic structure of genetics. From biological viewpoint, we defined various fundamental concepts andexamined their elementary properties that contain geneticsignificance. Mathematically, by genetic coalgebra, we mean anycoalgebra that occurs in genetics. They are generallynoncoassociative and without counit; and in the case ofnon-sex-linked inheritance, they are cocommutative. Eachcoalgebra with genetic realization has a baric property. We havealso discussed the methods to construct new genetic coalgebras,including cocommutative duplication, the tensor product, linearcombinations and the skew linear map, which allow us to describecomplex genetic traits. We also put forward certain theoremsthat state the relationship between gametic coalgebra and gameticalgebra. By Brower's theorem in topology, we prove the existenceof equilibrium state for the in-evolution operator. |
| format | Article |
| id | doaj-art-dd72099baaa64ffbb61eb105ac4169d4 |
| institution | Kabale University |
| issn | 1551-0018 |
| language | English |
| publishDate | 2004-06-01 |
| publisher | AIMS Press |
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| spelling | doaj-art-dd72099baaa64ffbb61eb105ac4169d42025-01-24T01:46:53ZengAIMS PressMathematical Biosciences and Engineering1551-00182004-06-011224326610.3934/mbe.2004.1.243Coalgebraic Structure of Genetic InheritanceJianjun Tian0Bai-Lian Li1Department of Mathematics, University of California, Riverside, CA 92521-0135Ecological Complexity and Modeling Laboratory, Department of Botany and Plant Sciences, University of California, Riverside, CA 92521-0124Although in the broadly defined genetic algebra, multiplicationsuggests a forward direction of from parents to progeny, when looking from the reversedirection, it also suggests to us a new algebraic structure ---coalgebraic structure, which we call genetic coalgebras. It isnot the dual coalgebraic structure and can be used in theconstruction of phylogenetic trees. Mathematically, to constructphylogenetic trees means we need to solve equations x[n]=a, or x(n)=b. It is generally impossible tosolve these equations in algebras. However, we can solve themin coalgebras in the sense of tracing back for their ancestors. Athorough exploration of coalgebraic structure in genetics isapparently necessary. Here, we develop a theoretical frameworkof the coalgebraic structure of genetics. From biological viewpoint, we defined various fundamental concepts andexamined their elementary properties that contain geneticsignificance. Mathematically, by genetic coalgebra, we mean anycoalgebra that occurs in genetics. They are generallynoncoassociative and without counit; and in the case ofnon-sex-linked inheritance, they are cocommutative. Eachcoalgebra with genetic realization has a baric property. We havealso discussed the methods to construct new genetic coalgebras,including cocommutative duplication, the tensor product, linearcombinations and the skew linear map, which allow us to describecomplex genetic traits. We also put forward certain theoremsthat state the relationship between gametic coalgebra and gameticalgebra. By Brower's theorem in topology, we prove the existenceof equilibrium state for the in-evolution operator.https://www.aimspress.com/article/doi/10.3934/mbe.2004.1.243general genetic algebrasbaric coalgebrasgeneral genetic coalgebrasin-evolution operators.conilpotent coalgebras |
| spellingShingle | Jianjun Tian Bai-Lian Li Coalgebraic Structure of Genetic Inheritance Mathematical Biosciences and Engineering general genetic algebras baric coalgebras general genetic coalgebras in-evolution operators. conilpotent coalgebras |
| title | Coalgebraic Structure of Genetic Inheritance |
| title_full | Coalgebraic Structure of Genetic Inheritance |
| title_fullStr | Coalgebraic Structure of Genetic Inheritance |
| title_full_unstemmed | Coalgebraic Structure of Genetic Inheritance |
| title_short | Coalgebraic Structure of Genetic Inheritance |
| title_sort | coalgebraic structure of genetic inheritance |
| topic | general genetic algebras baric coalgebras general genetic coalgebras in-evolution operators. conilpotent coalgebras |
| url | https://www.aimspress.com/article/doi/10.3934/mbe.2004.1.243 |
| work_keys_str_mv | AT jianjuntian coalgebraicstructureofgeneticinheritance AT bailianli coalgebraicstructureofgeneticinheritance |