Inverse Semigroup - The Natural Partial Order

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 ≤ t and st, then s = t. Similarly, if st.

On E(S), the natural partial order becomes:

so the product of any two idempotents in S is equal to the lesser of the two, with respect to ≤. If E(S) forms a chain (i.e., E(S) is totally ordered by ≤), then S is a union of groups.