Geodesic
Finite full transformation semigroups as collections of random functions
Glasgow mathematical journal, Tome 30 (1988) no. 2, pp. 203-211

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The collection of all self-maps on a non-empty set X under composition is known in algebraic semigroup theory as the full transformation semigroup on X and is written x. Its importance lies in the fact that any semigroup S can be embedded in the full transformation semigroup (where S1 is the semigroup S with identity 1 adjoined, if S does not already possess one). The proof is similar to Cayley's Theorem that a group G can be embedded in SG, the group of all bijections of G to itself. In this paper X will be a finite set of order n, which we take to be and so we shall write Tn for X. 
Brown, B.; Higgins, P. M. Finite full transformation semigroups as collections of random functions. Glasgow mathematical journal, Tome 30 (1988) no. 2, pp. 203-211. doi: 10.1017/S0017089500007230
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