Reduced Incidence Algebras
Any member of an incidence algebra that assigns the same value to any two intervals that are isomorphic to each other as posets is a member of the reduced incidence algebra. This is a subalgebra of the incidence algebra, and it clearly contains the incidence algebra's identity element and zeta function. Any element of the reduced incidence algebra that is invertible in the larger incidence algebra has its inverse in the reduced incidence algebra. As a consequence, the Möbius function is always a member of the reduced incidence algebra. Reduced incidence algebras shed light on the theory of generating functions, as alluded to in the case of the natural numbers above.
Read more about this topic: Incidence Algebra
Famous quotes containing the words reduced and/or incidence:
“Realism, whether it be socialist or not, falls short of reality. It shrinks it, attenuates it, falsifies it; it does not take into account our basic truths and our fundamental obsessions: love, death, astonishment. It presents man in a reduced and estranged perspective. Truth is in our dreams, in the imagination.”
—Eugène Ionesco (b. 1912)
“Hermann Goering, Joachim von Ribbentrop, Albert Speer, Walther Frank, Julius Streicher and Robert Ley did pass under my inspection and interrogation in 1945 but they only proved that National Socialism was a gangster interlude at a rather low order of mental capacity and with a surprisingly high incidence of alcoholism.”
—John Kenneth Galbraith (b. 1908)