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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

17D92 Genetic algebras
92D10 Genetics and epigenetics
Full Text: DOI
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