Geodesic
Length functions for semigroup embeddings
Algebra and discrete mathematics, Tome 9 (2010) no. 1, pp. 1-15.

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Following the work done in $[\mathrm O]$ for groups, we describe, for a given semigroup $S$, which functions $l\colon S\to\mathbb{N}$ can be realized up to equivalence as length functions $g\mapsto|g|_{H}$ by embedding $S$ into a finitely generated semigroup $H$. We also, following the work done in $[\mathrm O_2]$ and $[\mathrm{OS}]$, provide a complete description of length functions of a given finitely generated semigroup with enumerable set of relations inside a finitely presented semigroup.
Keywords: Membership problem, word problem, embeddings of semigroups, length function, distortion. 
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     author = {Tara Colleen Davis},
     title = {Length functions for semigroup embeddings},
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Tara Colleen Davis. Length functions for semigroup embeddings. Algebra and discrete mathematics, Tome 9 (2010) no. 1, pp. 1-15. http://geodesic.mathdoc.fr/item/ADM_2010_9_1_a1/