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A train algebra that is not special triangular. (English) Zbl 0616.17011
Two important classes of nonassociative algebras arising in population genetics are the train algebras and the special triangular (or Gonshor) algebras. Every special triangular algebra is train. The question of whether there exist commutative train algebras (over a field of characterization 0) that are not special triangular was open for 40 years before V. M. Abraham constructed a 6 dimensional example with 3 generators. He also showed that every train algebra of dimension $$\leq 4$$ is special triangular. I showed later that every train algebra with a single generator is special triangular.
The example of the present paper fills two gaps in that it is of dimension 5 and has 2 generators. Abraham’s example worked because it contains an ideal of codimension 1 which is nil but not nilopotent. The present case also shows that that property is not a necessary feature of a train, but non special triangular, algebra.

##### MSC:
 17D92 Genetic algebras 92D10 Genetics and epigenetics
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##### References:
 [1] V. M. Abraham, A note on train algebras. Proc. Edinburgh Math. Soc. (2)20, 53-58 (1976). · Zbl 0361.17007 · doi:10.1017/S0013091500015753 [2] I. M. H. Etherington, Genetic algebras. Proc. Royal Soc. Edinburgh59, 242-258 (1939). · Zbl 0027.29402 [3] H. Gonshor, Special train algebras arising in genetics. Proc. Edinburgh Math. Soc. (2)12, 41-53 (1960). · Zbl 0249.17003 · doi:10.1017/S0013091500025037 [4] H. Gonshor, Contributions to genetic algebras. Proc. Edinburgh Math. Soc. (2)17, 289-298 (1971). · Zbl 0247.92002 · doi:10.1017/S0013091500009548 [5] P. Holgate, Free non associative principal train algebras. Proc. Edinburgh Math. Soc. (2)26, 313-319 (1984). · Zbl 0555.17006 · doi:10.1017/S0013091500022458 [6] P.Holgate, Conditions for the linearisation of the squaring operator in genetic algebrasa review. In: Algèbres Génétiques. Ed. A. Micali, Paris, 23-34 (1987). [7] R. D. Schafer, Structure of genetic algebras. American J. Math.71, 121-135 (1949). · Zbl 0034.02004 · doi:10.2307/2372100 [8] D. Suttles, A counter example to a conjecture of Albert. Notices American Math. Soc. A19, 566 (1972). [9] A.Wörz-Busekros, Algebras in genetics. Lecture Notes in Biomathematics36, Berlin-Heidelberg-New York 1980. · Zbl 0431.92017
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