×

zbMATH — the first resource for mathematics

Free non-associative principal train algebras. (English) Zbl 0555.17006
The author extends the concepts of genetic algebra (which he calls special triangular) and train algebra to infinite dimensional algebras. He distinguishes between strongly special triangular and weakly special triangular. (In the finite dimensional case they are equivalent.) He then studies the free non-associative algebra \(F_ k\) on \(k\) generators. Theorem 1 says that \(F_ k\) is baric and weakly special triangular for all \(k\). Also \(F_k\) is strongly special triangular for \(k=1\) but not for \(k\geq 2\). He then defines the concept of free commutative nonassociative principal train algebra. He studies the question of when such algebras are finite. Finally he gives a method for linearizing the plenary operator in a specific algebra which is different from the general method of McHale and Ringwood. The latter method would lead to an infinite dimensional algebra, whereas the author’s method gives rise to an algebra of finite dimension.
Reviewer: Harry Gonshor

MSC:
17D92 Genetic algebras
92D10 Genetics and epigenetics
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] DOI: 10.1112/jlms/s1-42.1.489 · Zbl 0163.03103 · doi:10.1112/jlms/s1-42.1.489
[2] DOI: 10.1112/plms/s3-40.2.364 · Zbl 0393.17008 · doi:10.1112/plms/s3-40.2.364
[3] DOI: 10.1112/plms/s3-40.2.346 · Zbl 0388.17007 · doi:10.1112/plms/s3-40.2.346
[4] Abraham, Proc. Edinburgh Math. Soc. 20 pp 53– (1976)
[5] DOI: 10.1007/BF02450789 · Zbl 0387.92008 · doi:10.1007/BF02450789
[6] DOI: 10.1112/jlms/s2-28.1.17 · Zbl 0515.17010 · doi:10.1112/jlms/s2-28.1.17
[7] Holgate, J. Royal Statist. Soc. B43 pp 1– (1981)
[8] Holgate, Proc. Royal Soc. Edinburgh A 86 pp 65– (1980) · Zbl 0433.92015 · doi:10.1017/S0308210500012002
[9] Wörz-Busekros, Algebras in genetics (1980) · doi:10.1007/978-3-642-51038-0
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.