Genetic algebra

<h2 id="baric-algebras">Baric algebras</h2>
Baric algebras (or weighted algebras) were introduced by Etherington (1939). A baric algebra over a <a href="/facts/Field_(mathematics)/xAjAS4ko">field</a> K is a possibly non-associative algebra over K together with a homomorphism w, called the weight, from the algebra to K.<a class="footnote-ref" id="fnref:1" href="#fn:1">1</a>

<h2 id="bernstein-algebras">Bernstein algebras</h2>
A Bernstein algebra, based on the work of <a href="/facts/Sergei_Natanovich_Bernstein/m7aCQ2Y6">Sergei Natanovich Bernstein</a> (1923) on the <a href="/facts/Hardy%E2%80%93Weinberg_law/loy5NCKx">Hardy–Weinberg law</a> in genetics, is a (possibly non-associative) baric algebra B over a field K with a weight homomorphism w from B to K satisfying 
 
 
 
 (
 
 x
 
 2
 
 
 
 )
 
 2
 
 
 =
 w
 (
 x
 
 )
 
 2
 
 
 
 x
 
 2
 
 
 
 
 {\displaystyle (x^{2})^{2}=w(x)^{2}x^{2}}
 
. Every such algebra has idempotents e of the form 
 
 
 
 e
 =
 
 a
 
 2
 
 
 
 
 {\displaystyle e=a^{2}}
 
 with 
 
 
 
 w
 (
 a
 )
 =
 1
 
 
 {\displaystyle w(a)=1}
 
. The <a href="/facts/Peirce_decomposition/jF6H3lCx">Peirce decomposition</a> of B corresponding to e is

B
        =
        K
        e
        ⊕
        
          U
          
            e
          
        
        ⊕
        
          Z
          
            e
          
        
      
    
    {\displaystyle B=Ke\oplus U_{e}\oplus Z_{e}}

where 
 
 
 
 
 U
 
 e
 
 
 =
 {
 a
 ∈
 ker
 ⁡
 w
 :
 e
 a
 =
 a
 
 /
 
 2
 }
 
 
 {\displaystyle U_{e}=\{a\in \ker w:ea=a/2\}}
 
 and 
 
 
 
 
 Z
 
 e
 
 
 =
 {
 a
 ∈
 ker
 ⁡
 w
 :
 e
 a
 =
 0
 }
 
 
 {\displaystyle Z_{e}=\{a\in \ker w:ea=0\}}
 
. Although these subspaces depend on e, their dimensions are invariant and constitute the type of B. An exceptional Bernstein algebra is one with 
 
 
 
 
 U
 
 e
 
 
 2
 
 
 =
 0
 
 
 {\displaystyle U_{e}^{2}=0}
 
.<a class="footnote-ref" id="fnref:2" href="#fn:2">2</a>

<h2 id="copular-algebras">Copular algebras</h2>
Copular algebras were introduced by Etherington (1939, section 8)

<h2 id="evolution-algebras">Evolution algebras</h2>
An evolution algebra over a field is an algebra with a basis on which multiplication is defined by the product of distinct basis terms being zero and the square of each basis element being a linear form in basis elements. A real evolution algebra is one defined over the reals: it is non-negative if the structure constants in the linear form are all non-negative.<a class="footnote-ref" id="fnref:3" href="#fn:3">3</a> An evolution algebra is necessarily commutative and <a href="/facts/Flexible_algebra/JPpgC0kU">flexible</a> but not necessarily associative or <a href="/facts/Power-associative/192PkK5G">power-associative</a>.<a class="footnote-ref" id="fnref:4" href="#fn:4">4</a>

<h2 id="gametic-algebras">Gametic algebras</h2>
A gametic algebra is a finite-dimensional real algebra for which all structure constants lie between 0 and 1.<a class="footnote-ref" id="fnref:5" href="#fn:5">5</a>

<h2 id="genetic-algebras">Genetic algebras</h2>
Genetic algebras were introduced by Schafer (1949) who showed that special train algebras are genetic algebras and genetic algebras are train algebras.

<h2 id="special-train-algebras">Special train algebras</h2>
Special train algebras were introduced by Etherington (1939, section 4) as special cases of baric algebras.
A special train algebra is a baric algebra in which the kernel N of the weight function is nilpotent and the principal powers of N are ideals.<a class="footnote-ref" id="fnref:6" href="#fn:6">6</a>
Etherington (1941) showed that special train algebras are train algebras.

<h2 id="train-algebras">Train algebras</h2>
Train algebras were introduced by Etherington (1939, section 4) as special cases of baric algebras.
Let 
 
 
 
 
 c
 
 1
 
 
 ,
 …
 ,
 
 c
 
 n
 
 
 
 
 {\displaystyle c_{1},\ldots ,c_{n}}
 
 be elements of the field K with 
 
 
 
 1
 +
 
 c
 
 1
 
 
 +
 ⋯
 +
 
 c
 
 n
 
 
 =
 0
 
 
 {\displaystyle 1+c_{1}+\cdots +c_{n}=0}
 
. The formal polynomial

x
          
            n
          
        
        +
        
          c
          
            1
          
        
        w
        (
        x
        )
        
          x
          
            n
            −
            1
          
        
        +
        ⋯
        +
        
          c
          
            n
          
        
        w
        (
        x
        
          )
          
            n
          
        
      
    
    {\displaystyle x^{n}+c_{1}w(x)x^{n-1}+\cdots +c_{n}w(x)^{n}}

is a train polynomial. The baric algebra B with weight w is a train algebra if

a
          
            n
          
        
        +
        
          c
          
            1
          
        
        w
        (
        a
        )
        
          a
          
            n
            −
            1
          
        
        +
        ⋯
        +
        
          c
          
            n
          
        
        w
        (
        a
        
          )
          
            n
          
        
        =
        0
      
    
    {\displaystyle a^{n}+c_{1}w(a)a^{n-1}+\cdots +c_{n}w(a)^{n}=0}

for all elements 
 
 
 
 a
 ∈
 B
 
 
 {\displaystyle a\in B}
 
, with 
 
 
 
 
 a
 
 k
 
 
 
 
 {\displaystyle a^{k}}
 
 defined as principal powers, 
 
 
 
 (
 
 a
 
 k
 −
 1
 
 
 )
 a
 
 
 {\displaystyle (a^{k-1})a}
 
.<a class="footnote-ref" id="fnref:7" href="#fn:7">7</a><a class="footnote-ref" id="fnref:8" href="#fn:8">8</a>

<h2 id="zygotic-algebras">Zygotic algebras</h2>
Zygotic algebras were introduced by Etherington (1939, section 7)

<ul><li>Bernstein, S. N. (1923), "Principe de stationarité et généralisation de la loi de Mendel", C. R. Acad. Sci. Paris, 177: 581–584.</li>
<li>Bertrand, Monique (1966), Algèbres non associatives et algèbres génétiques, Mémorial des Sciences Mathématiques, Fasc. 162, Gauthier-Villars Éditeur, Paris, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0215885">0215885</a></li>
<li>Etherington, I. M. H. (1939), <a href="https://web.archive.org/web/20110706211658/http://math.usask.ca/~bremner/research/geneticalgebras/etherington/ga.pdf">"Genetic algebras"</a> (PDF), Proc. R. Soc. Edinburgh, 59: 242–258, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1017%2FS0370164600012323">10.1017/S0370164600012323</a>, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0000597">0000597</a>, <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:0027.29402">0027.29402</a>, archived from <a href="http://math.usask.ca/~bremner/research/geneticalgebras/etherington/ga.pdf">the original</a> (PDF) on 2011-07-06</li>
<li>Etherington, I. M. H. (1941), "Special train algebras", The Quarterly Journal of Mathematics, Second Series, 12: 1–8, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1093%2Fqmath%2Fos-12.1.1">10.1093/qmath/os-12.1.1</a>, <a href="/facts/ISSN_(identifier)/DPAflDvU">ISSN</a> <a href="https://search.worldcat.org/issn/0033-5606">0033-5606</a>, <a href="/facts/JFM_(identifier)/P6rFxKKx">JFM</a> <a href="https://zbmath.org/?format=complete&q=an:67.0093.04">67.0093.04</a>, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0005111">0005111</a>, <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:0027.29401">0027.29401</a></li>
<li>Lyubich, Yu.I. (2001) [1994], <a href="https://www.encyclopediaofmath.org/index.php?title=Bernstein_problem_in_mathematical_genetics&oldid=16709">"Bernstein problem in mathematical genetics"</a>, <a href="/facts/Encyclopedia_of_Mathematics/WC6mGtPm">Encyclopedia of Mathematics</a>, <a href="/facts/European_Mathematical_Society/B3h7b672">EMS Press</a></li>
<li>Micali, A. (2001) [1994], <a href="https://www.encyclopediaofmath.org/index.php?title=Baric_algebra&oldid=16628">"Baric algebra"</a>, <a href="/facts/Encyclopedia_of_Mathematics/WC6mGtPm">Encyclopedia of Mathematics</a>, <a href="/facts/European_Mathematical_Society/B3h7b672">EMS Press</a></li>
<li>Micali, A. (2001) [1994], <a href="https://www.encyclopediaofmath.org/index.php?title=Bernstein_algebra&oldid=11704">"Bernstein algebra"</a>, <a href="/facts/Encyclopedia_of_Mathematics/WC6mGtPm">Encyclopedia of Mathematics</a>, <a href="/facts/European_Mathematical_Society/B3h7b672">EMS Press</a></li>
<li>Reed, Mary Lynn (1997), "Algebraic structure of genetic inheritance", Bulletin of the American Mathematical Society, New Series, 34 (2): 107–130, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1090%2FS0273-0979-97-00712-X">10.1090/S0273-0979-97-00712-X</a>, <a href="/facts/ISSN_(identifier)/DPAflDvU">ISSN</a> <a href="https://search.worldcat.org/issn/0002-9904">0002-9904</a>, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=1414973">1414973</a>, <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:0876.17040">0876.17040</a></li>
<li>Schafer, Richard D. (1949), "Structure of genetic algebras", <a href="/facts/American_Journal_of_Mathematics/lkgftcD7">American Journal of Mathematics</a>, 71 (1): 121–135, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.2307%2F2372100">10.2307/2372100</a>, <a href="/facts/ISSN_(identifier)/DPAflDvU">ISSN</a> <a href="https://search.worldcat.org/issn/0002-9327">0002-9327</a>, <a href="/facts/JSTOR_(identifier)/YTeVmaJ7">JSTOR</a> <a href="https://www.jstor.org/stable/2372100">2372100</a>, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0027751">0027751</a></li>
<li>Tian, Jianjun Paul (2008), Evolution algebras and their applications, Lecture Notes in Mathematics, vol. 1921, Berlin: <a href="/facts/Springer-Verlag/nAesf6nT">Springer-Verlag</a>, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-3-540-74283-8, <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:1136.17001">1136.17001</a></li>
<li>Wörz-Busekros, Angelika (1980), Algebras in genetics, Lecture Notes in Biomathematics, vol. 36, Berlin, New York: <a href="/facts/Springer-Verlag/nAesf6nT">Springer-Verlag</a>, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-387-09978-1, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0599179">0599179</a></li>
<li>Wörz-Busekros, A. (2001) [1994], <a href="https://www.encyclopediaofmath.org/index.php?title=Genetic_algebra">"Genetic algebra"</a>, <a href="/facts/Encyclopedia_of_Mathematics/WC6mGtPm">Encyclopedia of Mathematics</a>, <a href="/facts/European_Mathematical_Society/B3h7b672">EMS Press</a></li></ul>
<h2 id="further-reading">Further reading</h2>
<ul><li>Lyubich, Yu.I. (1983), Mathematical structures in population genetics. (Matematicheskie struktury v populyatsionnoj genetike) (in Russian), Kiev: Naukova Dumka, <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:0593.92011">0593.92011</a></li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">González, S.; Martínez, C. (2001), "About Bernstein algebras", in Granja, Ángel (ed.), Ring theory and algebraic geometry. Proceedings of the 5th international conference on algebra and algebraic geometry, SAGA V, León, Spain, Lect. Notes Pure Appl. Math., vol. 221, New York, NY: Marcel Dekker, pp. 223–239, Zbl 1005.17021 <a href="/wiki/Zbl_(identifier)" target="_blank">/wiki/Zbl_(identifier)</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">Catalan, A. (2000). "E-ideals in Bernstein algebras". In Costa, Roberto (ed.). Nonassociative algebra and its applications. Proceedings of the fourth international conference, São Paulo, Brazil. Lect. Notes Pure Appl. Math. Vol. 211. New York, NY: Marcel Dekker. pp. 35–42. Zbl 0968.17013. <a href="/wiki/Zbl_(identifier)" target="_blank">/wiki/Zbl_(identifier)</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">Tian (2008) p.18 <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">Tian (2008) p.20 <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn:5">Cohn, Paul M. (2000). Introduction to Ring Theory. Springer Undergraduate Mathematics Series. Springer-Verlag. p. 56. ISBN 1852332069. ISSN 1615-2085. <a href="1852332069" target="_blank">1852332069</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
<li id="fn:6">González, S.; Martínez, C. (2001), "About Bernstein algebras", in Granja, Ángel (ed.), Ring theory and algebraic geometry. Proceedings of the 5th international conference on algebra and algebraic geometry, SAGA V, León, Spain, Lect. Notes Pure Appl. Math., vol. 221, New York, NY: Marcel Dekker, pp. 223–239, Zbl 1005.17021 <a href="/wiki/Zbl_(identifier)" target="_blank">/wiki/Zbl_(identifier)</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></li>
<li id="fn:7">González, S.; Martínez, C. (2001), "About Bernstein algebras", in Granja, Ángel (ed.), Ring theory and algebraic geometry. Proceedings of the 5th international conference on algebra and algebraic geometry, SAGA V, León, Spain, Lect. Notes Pure Appl. Math., vol. 221, New York, NY: Marcel Dekker, pp. 223–239, Zbl 1005.17021 <a href="/wiki/Zbl_(identifier)" target="_blank">/wiki/Zbl_(identifier)</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></li>
<li id="fn:8">Catalán S., Abdón (1994). "E-ideals in baric algebras". Mat. Contemp. 6: 7–12. Zbl 0868.17023. <a href="/wiki/Zbl_(identifier)" target="_blank">/wiki/Zbl_(identifier)</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></li>
</ol>

Genetic algebra open-in-new

Genetic algebra