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 partial order, natural, partial and/or order:
“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)
“Peter: I’ll bet there isn’t a good piggy-back rider in your whole family. I never knew a rich man yet who could piggy-back ride.
Ellie: You’re prejudiced.
Peter: You show me a good piggy-backer and I’ll show you a real human. Now you take Abraham Lincoln, for instance. A natural piggy-backer.”
—Robert Riskin (1897–1955)
“The one-eyed man will be King in the country of the blind only if he arrives there in full possession of his partial faculties—that is, providing he is perfectly aware of the precise nature of sight and does not confuse it with second sight ... nor with madness.”
—Angela Carter (1940–1992)
“You cannot have one well-bred man without a whole society of such. They keep each other up to any high point. Especially women;Mit requires a great many cultivated women,—saloons of bright, elegant, reading women, accustomed to ease and refinement, to spectacles, pictures, sculpture, poetry, and to elegant society, in order that you have one Madame de Staël.”
—Ralph Waldo Emerson (1803–1882)