Topology and extensions of Mendelian inheritance

Subject Areas : General algebraic systems

H. Mirabdollahi ¹ (Department of Mathematical Sciences, Yazd University, Yazd, Iran)
B. Davvaz ² (Department of Mathematical Sciences, Yazd University, Yazd, Iran)

Received: 2023-11-29 Accepted : 2024-01-25 Published : 2024-02-14

Keywords: Impact, Influence, Discrete topology, indiscrete topology, dependence relation, alleles of a gene, complete dominant, incomplete dominant, codominant,

Abstract :

In this paper, we seek to provide an appropriate topology for a set of alleles of a gene that have arbitrary interactions between its alleles. For this purpose, we first define a suitable topology on a set of alleles between which there is a similar interaction. Next, with the help of the topology on the set of alleles of a gene, we equip the set of phenotypes of the desired gene with a suitable topology. In addition, the final section will discuss the dependence relations between the alleles of a gene.‎

References:

[1] M. Al Tahan, B. Davvaz, Algebraic hyperstructures associated to biological inheritance, Math. Biosci. 285 (2017), 112-118.

[2] M. Al Tahan, B. Davvaz, N-ary hyperstructures associated to the genotypes of F2-Offspring, Inter. J. Biomath. 10 (2017), 8:1750118.

[3] R. Ameri, Topological (transposition) hypergroups, Ital. J. Pure. Appl. Math. 13 (2003), 181-186.

[4] J. Chvalina, Functional Graphs, Quasi-Ordered Sets and Commutative Hypergroups, Masaryk University, Brno (in Czech), 1995.

[5] I. Cristea, J. Kocijan, M. Nov´ ak, Introduction to dependence relations and their links to algebraic hyperstructures, Mathematics. 7 (2019), 10:885.

[6] P. Corsini, Prolegomena of Hypergroup Theory, 2nd edition, Aviani editor, 1993.

[7] P. Corsini, V. Leoreanu, Applications of Hyperstructures Theory, Advanced in Mathematics, Kluwer Academic Publisher, 2003.

[8] P. Corsini, V. Leoreanu, Applications of Hyperstructure Theory, Kluwer, Dordrecht, 2003.

[9] C. Darwin, The Variation of Animals and Plants Under Domestication (1st American edition), Orange Judd and Co., New York, 1868.

[10] B. Davvaz, A Brief Survey of the Theory of hv-Structures, Algebraic Hyper-structures and Applications, Spanidis, Xanthi, 2003.

[11] B. Davvaz, Polygroup Theory and Related System, World Scientific Publishing, 2013.

[12] B. Davvaz, A. Dehghan Nezhad, M. M. Heidari, Inheritance examples of algebraic hyperstructures, Inform. Sci. 224 (2013), 180-187.

[13] B. Davvaz, V. Leoreanu-Fotea, Hyperring Theory and Applications, International Academic Press, 115, Palm Harber, USA, 2007.

[14] M. Ghadiri, B. Davvaz, Hv-Semigroup structure on F2-offspring of a gene pool, Inter. J. Biomath. 5 (2012), 4:1250011.

[15] S. Hoskova-Mayerova, An overview of topological and fuzzy topological hypergroupoids, Ratio. Math. 33 (2017), 21-38.

[16] S. Hoskova-Mayerova, Topological hypergroupoids, Comput. Math. Appl. 64 (2012), 2845-2849.

[17] F. Marty, Sur une Generalization de la Notion de Groupe, 8th Congress Math. Scandenaves, Stockholm, 1934.

[18] P. Rackova, Hypergroups of Symmetric Matrices, 10th International Congress of Algebraic Hyperstructures and Applications, Proceeding of AHA, 2008.

[19] I. G. Rosenberg, Hypergroups and join spaces determined by relations, Ital. J. Pure. Appl. Math. 4 (1998), 93-101.

[20] T. Vougiouklis, Hyperstructures and Their Representations, Hadronic Press, Palm Harbour, 1994.

[21] T. Vougiouklis, Generalization of P-hypergroups, Rend. Circ. Mat. Palermo. 36 (II) (1987), 114-121.

[22] T. Vougiouklis, On some representations of hypergroups, Ann. Sci. Univ. Clermont-Ferrand II Math. 26 (1990), 21-29.

[23] T. Vougiouklis, Representation of hypergroups by generalized permutations, Algeb. Univ. 29 (2) (1992), 172-183.