Editing Semigroup (section)

== Group of fractions ==
The '''group of fractions''' or '''group completion''' of a semigroup ''S'' is the [[group (mathematics)|group]] {{nowrap|1=''G'' = ''G''(''S'')}} generated by the elements of ''S'' as generators and all equations {{nowrap|1=''xy'' = ''z''}} that hold true in ''S'' as [[presentation of a group|relations]].<ref>{{Cite book|first=B. |last=Farb |title=Problems on mapping class groups and related topics |publisher=Amer. Math. Soc. |year=2006 |isbn=978-0-8218-3838-9 |page=357 }}</ref>  There is an obvious semigroup homomorphism {{nowrap|''j'' : ''S'' → ''G''(''S'')}} that sends each element of ''S'' to the corresponding generator. This has a [[universal property]] for morphisms from ''S'' to a group:<ref>{{Cite book|first1=M. |last1=Auslander |first2=D. A. |last2=Buchsbaum |title=Groups, rings, modules |publisher=Harper & Row |year=1974 |isbn=978-0-06-040387-4 |page=50 }}</ref>  given any group ''H'' and any semigroup homomorphism {{nowrap|''k'' : ''S'' → ''H''}}, there exists a unique [[group homomorphism]] {{nowrap|''f'' : ''G'' → ''H''}} with {{nowrap|1=''k'' = ''fj''}}. We may think of ''G'' as the "most general" group that contains a homomorphic image of ''S''.

An important question is to characterize those semigroups for which this map is an embedding.  This need not always be the case: for example, take ''S'' to be the semigroup of subsets of some set ''X'' with [[set-theoretic intersection]] as the binary operation (this is an example of a semilattice).  Since {{nowrap|1=''A''.''A'' = ''A''}} holds for all elements of ''S'', this must be true for all generators of ''G''(''S'') as well, which is therefore the [[trivial group]].  It is clearly necessary for embeddability that ''S'' have the [[cancellation property]].  When ''S'' is commutative this condition is also sufficient{{sfn|ps=|Clifford|Preston|1961|p=34}} and the [[Grothendieck group]] of the semigroup provides a construction of the group of fractions.  The problem for non-commutative semigroups can be traced to the first substantial paper on semigroups.{{sfn|ps=|Suschkewitsch|1928}}<ref>{{Cite book|url=http://www.gap-system.org/~history/Extras/Preston_semigroups.html|title=Personal reminiscences of the early history of semigroups|first=G. B.|last=Preston|year=1990|access-date=2009-05-12|author-link=Gordon Preston|url-status=dead|archive-url=https://web.archive.org/web/20090109045100/http://www.gap-system.org/~history/Extras/Preston_semigroups.html|archive-date=2009-01-09}}</ref>  [[Anatoly Maltsev]] gave necessary and sufficient conditions for embeddability in 1937.<ref>{{Cite journal| doi=10.1007/BF01571659 | last=Maltsev | first=A. | author-link=Anatoly Maltsev | title=On the immersion of an algebraic ring into a field | journal=Math. Annalen | volume=113 | year=1937 | pages=686–691 | s2cid=122295935 }}</ref>