Geodesic
Digraphs and the semigroup of all functions on a finite set
Glasgow mathematical journal, Tome 30 (1988) no. 1, pp. 41-57

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Let Tn denote the full transformation semigroup on the finite set = {1, 2, ... n}, that is the set of all mappings from to , with function composition as the semigroup operation. In this paper algorithms are introduced to solve equations such as axmb = c and ax = xb (a, b, c ∊ Tn), which employ a representation of members of Tn as special directed graphs. 
Higgins, Peter M. Digraphs and the semigroup of all functions on a finite set. Glasgow mathematical journal, Tome 30 (1988) no. 1, pp. 41-57. doi: 10.1017/S0017089500007011
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