A Representation Theorem for Co-diagonalizable Algebras

This abstract is a summary of a lecture given at the Seminar of the Department of Logic and Methodology of Sciences, Wroc law University, 23 May 1985. The present work refers directly to the investigations of Buszkowski and Prucnal [1] and that of Esakia [2], generalizing their results. Our main representation theorem for co-diagonalizable algebras (Theorem 2) is obtained by application of certain methods taken from Jónsson-Tarski [3]. Definition 1. By a derivative algebra (D-algebra in short) we shall mean a system such that (i) is a Boolean algebra, (ii) d is a unary operation on A satisfying the following conditions: a) d(0) = 0, b) d(a ∨ b) = d(a) ∨ d(b), c) d(d(a)) ≤ d(a), for all a, b ∈ A. Definition 2. We say that an algebra M = < A,∧,∨,−, 0, d > is a co-diagonalizable algebra (CD-algebra in short) if M is a D-algebra and, moreover, the operation d satisfies the following Löb’s condition: d(a) = d(a− d(a)). Co-diagonalizable algebras are called by some writers (e.g. Esakia [2]) Magari algebras. A Representation Theorem for Co-Diagonalizable Algebras 125 Typical examples of D-algebras and CD-algebras. 1. Let X be a topological space and let, for any A ⊆ X, d(A) be the set of all accumulation points of A (i.e. d(A) is the derivative of A). Then < P (X),∩,∪,−, ∅, d > is a D-algebra. It is called the derivative algebra over X. 2. Let X be a scattered topological space, i.e., no non-empty subset of X is dense-in-itself. Then is an CD-algebra (with d defined as above). This algebra is called the CD-algebra over X (cf. [2]). Theorem 1. Every derivative algebra is isomorphic with a subalgebra of the derivative algebra over a topological space. Theorem 2. Every CD-algebra is isomorphic with a subalgebra of the CD-algebra over a scattered topological space.