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)
“If, in looking at the lives of princes, courtiers, men of rank and fashion, we must perforce depict them as idle, profligate, and criminal, we must make allowances for the rich men’s failings, and recollect that we, too, were very likely indolent and voluptuous, had we no motive for work, a mortal’s natural taste for pleasure, and the daily temptation of a large income. What could a great peer, with a great castle and park, and a great fortune, do but be splendid and idle?”
—William Makepeace Thackeray (1811–1863)
“There is no luck in literary reputation. They who make up the final verdict upon every book are not the partial and noisy readers of the hour when it appears; but a court as of angels, a public not to be bribed, not to be entreated, and not to be overawed, decides upon every man’s title to fame. Only those books come down which deserve to last.”
—Ralph Waldo Emerson (1803–1882)
“The principles of the good society call for a concern with an order of being—which cannot be proved existentially to the sense organs—where it matters supremely that the human person is inviolable, that reason shall regulate the will, that truth shall prevail over error.”
—Walter Lippmann (1889–1974)