written x−1. Inverses in an inverse semigroup have many of the same properties as inverses in a group, for example, (ab)−1 = b−1a−1. In an inverse monoid, xx−1 and x−1x are not necessarily equal to the identity, but they are both idempotent. An inverse monoid S in which xx−1 = 1 = x−1x, for all x in S (a unipotent inverse monoid), is, of course, a group. There are a number of equivalent characterisations of an inverse semigroup S: Every element of S has a unique inverse, in the above sense. Every element of S has at least one inverse (S is a regular semigroup) and idempotents commute (that is, the idempotents of S form a semilattice). Every -class and every -class contains precisely one idempotent, where and are two of Green's relations. The idempotent in the -class of s is s−1s, whilst the idempotent in the -class of s is ss−1. There is therefore a simple characterisation of Green's relations in an inverse semigroup: Unless stated otherwise, E(S) will denote the semilattice of idempotents of an inverse semigroup S. Examples of inverse semigroups Partial bijections on a set X form an inverse semigroup under composition. Every group is an inverse semigroup. The bicyclic semigroup is inverse, with (a,b)−1 = (b,a). Every semilattice is inverse. The Brandt semigroup is inverse. The Munn semigroup is inverse. Multiplication table example. It is associative and every element has its own inverse according to aba = a, bab = b. It has no identity and is not commutative. The natural partial order An inverse semigroup S possesses a natural partial order relation ≤ (sometimes denoted by ω), which is defined by the following: for some idempotent e in S. Equivalently, for some (in general, different) idempotent f in S. In fact, e can be taken to be aa−1 and f to be a−1a. The natural partial order is compatible with both multiplication and inversion, that is, and In a group, this partial order simply reduces to equality, since the identity is the only idempotent. In a symmetric inverse semigroup, the partial order reduces to restriction of mappings, i.e., α ≤ β if, and only if, the domain of α is contained in the domain of β and xα = xβ, for all x in the domain of α. The natural partial order on an inverse semigroup interacts with Green's relations as follows: if s