and β are composed if, and only if, the image of α is equal to the domain of β; otherwise, the composition αβ is undefined. Under this alternative composition, the collection of all partial one-one transformations of a set forms not an inverse semigroup but an inductive groupoid, in the sense of category theory. This close connection between inverse semigroups and inductive groupoids is embodied in the Ehresmann–Schein–Nambooripad Theorem, which states that an inductive groupoid can always be constructed from an inverse semigroup, and conversely. More precisely, an inverse semigroup is precisely a groupoid in the category of posets that is an étale groupoid with respect to its (dual) Alexandrov topology and whose poset of objects is a meet-semilattice. Generalisations of inverse semigroups As noted above, an inverse semigroup S can be defined by the conditions (1) S is a regular semigroup, and (2) the idempotents in S commute; this has led to two distinct classes of generalisations of an inverse semigroup: semigroups in which (1) holds, but (2) does not, and vice versa. Examples of regular generalisations of an inverse semigroup are: Regular semigroups: a semigroup S is regular if every element has at least one inverse; equivalently, for each a in S, there is an x in S such that axa = a. Locally inverse semigroups: a regular semigroup S is locally inverse if eSe is an inverse semigroup, for each idempotent e. Orthodox semigroups: a regular semigroup S is orthodox if its subset of idempotents forms a subsemigroup. Generalised inverse semigroups: a regular semigroup S is called a generalised inverse semigroup if its idempotents form a normal band, i.e., xyzx = xzyx, for all idempotents x, y, z. The class of generalised inverse semigroups is the intersection of the class of locally inverse semigroups and the class of orthodox semigroups. Amongst the non-regular generalisations of an inverse semigroup are: (Left, right, two-sided) adequate semigroups. (Left, right, two-sided) ample semigroups. (Left, right, two-sided) semiadequate semigroups. Weakly (left, right, two-sided) ample semigroups. Inverse category This notion of inverse also readily generalizes to categories. An inverse category is simply a category in which every morphism has a generalized inverse such that and . An inverse category is selfdual. The category of sets and partial bijections is the prime example. Inverse categories have found various applications in theoretical computer science. See also Notes References English translation(PDF) Further reading For a brief