📖 Partial orders on a C-algebra..

The study different partial orders on a C-algebra and their properties. First we define a partial ordering on C-algebra by x ? y if and only if y ? x = x and we study the properties of this partial ordering. We give a number of equivalent conditions in terms of this partial ordering for a C-algebra to become a Boolean Algebra. Using the decomposition theorem we prove that for any a, b ? B(A) with a ? b = F, Aa is isomorphic to Ab if and only if there exists an isomorphism on A which sends a to b. We define two binary operations ? and ? on a C-algebra A and prove that < A, ? >and < A, ? > are semi lattices. We derive some properties of the partial orderings ??and ?? induced from the semilattices.