Hostname: page-component-586b7cd67f-vdxz6 Total loading time: 0 Render date: 2024-11-25T04:07:20.858Z Has data issue: false hasContentIssue false
  • English
  • Français

ON THE ALGEBRAIC PROPERTIES OF THE HUMAN ABO-BLOOD GROUP INHERITANCE PATTERN

Part of: General nonassociative rings Other nonassociative rings and algebras

Published online by Cambridge University Press:  21 July 2016

J. M. CASAS Open the ORCID record for J. M. CASAS [Opens in a new window] ,
M. LADRA Open the ORCID record for M. LADRA [Opens in a new window] ,
B. A. OMIROV Open the ORCID record for B. A. OMIROV [Opens in a new window]  and
R. TURDIBAEV Open the ORCID record for R. TURDIBAEV [Opens in a new window]
J. M. CASAS
Affiliation:
Dpto. Matemática Aplicada I, Universidad de Vigo, E. E. Forestal, Campus Universitario A Xunqueira, 36005 Pontevedra, Spain email jmcasas@uvigo.es
M. LADRA*
Affiliation:
Department of Algebra, University of Santiago de Compostela15782, Spain email manuel.ladra@usc.es, rustamtm@yahoo.com
B. A. OMIROV
Affiliation:
Institute of Mathematics, National University of Uzbekistan, Tashkent, 100125, Uzbekistan email omirovb@mail.ru
R. TURDIBAEV
Affiliation:
Department of Algebra, University of Santiago de Compostela15782, Spain email manuel.ladra@usc.es, rustamtm@yahoo.com
*
✉manuel.ladra@usc.es
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We generate an algebra on blood phenotypes with multiplication based on the human ABO-blood group inheritance pattern. We assume that gametes are not chosen randomly during meiosis. We investigate some of the properties of this algebra, namely, the set of idempotents, lattice of ideals and the associative enveloping algebra.

Keywords

Human ABO-blood group inheritanceblood phenotypealgebraabsolute nilpotent elementidempotentsideallatticeassociative algebra

MSC classification

Primary: 17D92: Genetic algebras
Secondary: 17A20: Flexible algebras 92D25: Population dynamics (general)
Type
Research Article
Information
The ANZIAM Journal , Volume 58 , Issue 1 , July 2016 , pp. 78 - 95
Copyright
© 2016 Australian Mathematical Society 

References

Bernstein, F., “Über die Erblichkeit der Blutgruppen”, Z. f. induktive Abstammungslehre 54 (1930) 400–426; doi:10.1007/BF01848967.Google Scholar
Bernstein, S. N., “Solution of a mathematical problem connected with the theory of heredity”, Ann. Math. Statistics 13 (1942) 53–61; doi:10.1214/aoms/1177731642.Google Scholar
Chakraborty, R., “Gene frequency estimates in the $\text{ABO}$ system and their efficiencies”, Sankhyā Ser. B 32 (1970) 21–26; http://www.jstor.org/stable/25051715.Google Scholar
Fujita, Y., Tanimura, M. and Tanaka, K., “The distribution of the $\text{ABO}$ blood groups in japan”, Jap. J. Human. Genet. 23 (1978) 63–109; doi:10.1007/BF02001790.Google Scholar
Ganikhodjaev, N., Daoud, J. I. and Usmanova, M., “Linear and nonlinear models of heredity for blood groups and Rhesus factor”, J. Appl. Sci. 10 (2010) 1748–1754; doi:10.3923/jas.2010.1748.1754.CrossRefGoogle Scholar
Ganikhodjaev, N., Saburov, M. and Jamilov, U., “Mendelian and non-Mendelian quadratic operators”, Appl. Math. Inf. Sci. 7 (2013) 1721–1729; doi:10.12785/amis/070509.Google Scholar
Greenwood, S. R. and Seber, G. A. F., “Estimating blood phenotype probabilities and their products”, Biometrics 48 (1992) 143–154; doi:10.2307/2532745.Google Scholar
Kang, S. H., Fukumori, Y., Ohnoki, S., Shibata, H., Han, K. S., Nishimukai, H. and Okubo, Y., “Distribution of abo genotypes and allele frequencies in a Korean population”, Jap. J. Human. Genet. 42 (1997) 331–335; doi:10.1007/BF02766955.Google Scholar
Landsteiner, K., “Zur Kenntnis der antifermentativen, lytischen und agglutinierenden Wirkungen des Blutserums und der Lymphe”, Zbl. Bakt. 27 (1900) 357–362.Google Scholar
Lyubich, Y. I., “Basic concepts and theorems of the evolutionary genetics of free populations”, Russian Math. Surv. 26 (1971) 51–123; doi:10.1070/RM1971v026n05ABEH003829.Google Scholar
Lyubich, Y. I., “Mathematical structures in population genetics”, in: Biomathematics, Volume 22 (Springer, Berlin, 1992).Google Scholar
Sadykov, T., “Polynomial dynamics of human blood genotypes frequencies”, J. Symbolic Comput. (2016); doi:10.1016/j.jsc.2016.02.012 (online).Google Scholar
Yamaguchi, H., Okuba, Y. and Hazama, F., “An $\text{A}_{2}\text{B}_{3}$ phenotype blood showing atypical mode of inheritance”, Proc. Japan Acad. 41 (1965) 316–320; doi:10.2183/pjab1945.41.316.Google Scholar
Yamaguchi, H., Okuba, Y. and Hazama, F., “Another japanese $\text{A}_{2}\text{B}_{3}$ blood-group family with propositus having O-group father”, Proc. Japan Acad. 42 (1966) 517–520; doi:10.2183/pjab1945.42.517.Google Scholar