, A ≤ B if, and only if, gA ≤ gB; for all g, h in G and all A in , g(hA) = (gh)A. The triple is also assumed to have the following properties: for every X in , there exists a g in G and an A in such that gA = X; for all g in G, g and have nonempty intersection. Such a triple is called a McAlister triple. A McAlister triple is used to define the following: together with multiplication . Then is an inverse semigroup under this multiplication, with (A,g)−1 = (g−1A, g−1). One of the main results in the study of E-unitary inverse semigroups is McAlister's P-Theorem: McAlister's P-Theorem. Let be a McAlister triple. Then is an E-unitary inverse semigroup. Conversely, every E-unitary inverse semigroup is isomorphic to one of this type. F-inverse semigroups An inverse semigroup is said to be F-inverse if every element has a unique maximal element above it in the natural partial order, i.e. every σ-class has a maximal element. Every F-inverse semigroup is an E-unitary monoid. McAlister's covering theorem has been refined by M.V. Lawson to: Theorem. Every inverse semigroup has an F-inverse cover. McAlister's P-theorem has been used to characterize F-inverse semigroups as well. A McAlister triple is an F-inverse semigroup if and only if is a principal ideal of and is a semilattice. 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. Connections with category theory The above composition of partial transformations of a set gives rise to a symmetric inverse semigroup. There is another way of composing partial transformations, which is more restrictive than that used above: two partial transformations α