Decompositions of a C-algebra

We prove that if A is a C-algebra, then for each a ∈ A, Aa = {x ∈ A/x ≤ a} is itself a C-algebra and is isomorphic to the quotient algebra A/θa of A where θa = {(x, y) ∈ A×A/a∧ x = a∧ y}. If A is C-algebra with T , we prove that for every a ∈ B(A), the centre of A, A is isomorphic to Aa ×Aa′ and that if A is isomorphic A1 ×A2, then there exists a∈ B(A) such that A1 is isomorphic Aa and A2 is isomorphic to Aa′ . Using this decomposition theorem, we prove that if a,b ∈ B(A) with a∧ b = F, then Aa is isomorphic to Ab if and only if there exists an isomorphism φ on A such that φ (a)= b.