Geodesic
Square roots in finite full transformation semigroups
Glasgow mathematical journal, Tome 23 (1982) no. 2, pp. 137-149

Voir la notice de l'article provenant de la source Cambridge University Press

Let X be a finite set and let (X) be the full transformation semigroup on X, i.e. the set of all mappings from X into X, the semigroup operation being composition of mappings. This paper aims to characterize those elements of (X) which have square roots. An easily verifiable necessary condition, that of being quasi-square, is found in Theorem 2, and in Theorems 4 and 5 we find necessary and sufficient conditions for certain special elements of (X). The property of being compatibly amenable is shown in Theorem 7 to be equivalent for all elements of (X) to the possession of a square root. 
Snowden, Mary; Howie, J. M. Square roots in finite full transformation semigroups. Glasgow mathematical journal, Tome 23 (1982) no. 2, pp. 137-149. doi: 10.1017/S0017089500004912
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