Free Inverse Semigroups
A construction similar to a free group is possible for inverse semigroups. A presentation of the free inverse semigroup on a set X may be obtained by considering the free semigroup with involution, where involution is the taking of the inverse, and then taking the quotient by the Vagner congruence
The word problem for free inverse semigroups is much more intricate than that of free groups. A celebrated result in this area due to W. D. Munn who showed that elements of the free inverse semigroup can be naturally regarded as trees, known as Munn trees. Multiplication in the free inverse semigroup has a correspondent on Munn trees, which essentially consists of overlapping common portions of the trees. (see Lawson 1998 for further details)
Any free inverse semigroup is F-inverse.
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