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Nonadmissible genealogical trees. (English) Zbl 0741.05022

The author continues her study of admisible words initiated in a previous paper [same journal 75, 247-251 (1986; Zbl 0602.16026)]. Admissiblity of a word is defined in terms of admissibility of an oriented tree, called a genealogical tree. The author obtains a criterion for the nonadmissibility of a genealogical tree, and uses it to determine a class of nonadmissible words. These results are applied to investigate the structure of a module constructed over a particularly simple genealogical tree which is shown to provide an example of an injective cogenerator for the category of all \(K[x]\)-modules (where \(K[x]\) denotes the polynomial algebra over a field \(K\)).

MSC:

05C05 Trees
05C25 Graphs and abstract algebra (groups, rings, fields, etc.)

Citations:

Zbl 0602.16026
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References:

[1] G. D’Este , Extended strings and admissible words , Rend. Sem. Mat. Univ. Padova , 75 ( 1986 ), pp. 247 - 251 . Numdam | MR 847669 | Zbl 0602.16026 · Zbl 0602.16026
[2] C.M. Ringel , The indecomposable representations of the dihedral 2-groups , Math. Ann. , 214 ( 1975 ), pp. 19 - 43 . MR 364426 | Zbl 0299.20005 · Zbl 0299.20005 · doi:10.1007/BF01428252
[3] D.W. Sharpe - P. Vamos , Injective modules , Cambridge University Press , 1972 . MR 360706 | Zbl 0245.13001 · Zbl 0245.13001
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