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.
Read more about this topic: Inverse Semigroup
Famous quotes containing the words natural, partial and/or order:
“The open frontier, the hardships of homesteading from scratch, the wealth of natural resources, the whole vast challenge of a continent waiting to be exploited, combined to produce a prevailing materialism and an American drive bent as much, if not more, on money, property, and power than was true of the Old World from which we had fled.”
—Barbara Tuchman (1912–1989)
“Both the man of science and the man of art live always at the edge of mystery, surrounded by it. Both, as a measure of their creation, have always had to do with the harmonization of what is new with what is familiar, with the balance between novelty and synthesis, with the struggle to make partial order in total chaos.... This cannot be an easy life.”
—J. Robert Oppenheimer (1904–1967)
“People are not the way they are primarily in order to annoy me.”
—Mason Cooley (b. 1927)