Hanna Neumann in 


[American Journal of Mathematics, 1948, 


<http://www.jstor.org/discover/10.2307/2372201?uid=2&uid=4&sid=21102497379451> ] 


introduced a notion of generalized free product of groups $G\_i$ with amalgamated subgroups $H\_{ij}< G\_i$. Did anybody consider this construction for semigroups? Thank you in advance. 


This question arose while studying partial actions of groups on semigroups.



Generalized free product of semigroups with amalgamated subsemigroups
OK, here is my comment expanded to an answer. I looked at Hanna Neumann's definition again. What she defined is *not* a special case graph of groups, but of what is now called (again, in a special case) the "fundamental group of a complex of groups". Indeed, the modern definition of the latter is category-theoretic one (the limit of a commutative diagram of groups), details could be found for instance in the book by Bridson and Haefliger "Metric Spaces of Non-Positive Curvature". This definition extends verbatim to the case of semigroups. It is then well-known that vertex groups in such complex do not (in general) embed in the limit. However, they do provided that the complex is *developable*, which, in turn, is implied for instance by the local CAT(0) assumption. 


Search for "complex of semigroups" returned nothing. My guess then is that you are on your own. My suggestion is to look for a CAT(0)-concept which might work in this context and would yield a "developable complex" of semigroups and guarantee, say, that vertex semigroups are embedded. Reading first Stallings' paper "Triangles of groups" (where "angles" are defined purely algebraically) might be a good start. See also [here](http://www.ams.org/journals/tran/1995-347-11/S0002-9947-1995-1316847-6/S0002-9947-1995-1316847-6.pdf) and Bridson-Haefliger. 


