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)
“PLAYING SHOULD BE FUN! In our great eagerness to teach our children we studiously look for “educational” toys, games with built-in lessons, books with a “message.” Often these “tools” are less interesting and stimulating than the child’s natural curiosity and playfulness. Play is by its very nature educational. And it should be pleasurable. When the fun goes out of play, most often so does the learning.”
—Joanne E. Oppenheim (20th century)
“You must not be partial in judging: hear out the small and the great alike; you shall not be intimidated by anyone, for the judgment is God’s.”
—Bible: Hebrew, Deuteronomy 1:17.
“It is the international system of currency which determines the totality of life on this planet. That is the natural order of things today. That is the atomic, and sub-atomic, and galactic
structure of things today. And you have meddled with the primal forces of nature! And you will atone! Am I getting through to you, Mr. Beale?”
—Paddy Chayefsky (1923–1981)