Closure algebras

• Main Author: Logan, G. J.
• Language: en
• Published: University of Canterbury. Mathematics 2013
• Online Access: http://hdl.handle.net/10092/8418

This thesis investigates the properties and behaviour of closure algebras. Closure algebras generalize the concepts of topological closure, algebraic closure and logical consequence. A closure algebra consists of a unary function C, defined on the power set P (X) of a given set X, and satisfying three axioms A ⊆ C(A), C(C(A)) = C(A), A ⊆ B ⇒ C(A) ⊆ C(B), for each A,B ∈ P (X). These structures are considered along with their dual spaces; topological spaces constructed by generalizing the methods of M.H. Stone in his work on the representations of boolean algebras. A representation theorem for T₁ spaces is obtained. The notions of subalgebra, homomorphism and congruence are defined for closure algebras, so that the definitions generalize standard usage, and enable analogues of some of the major theorems of Universal Algebra to be proved. Using these definitions and the definition of a closure product, it becomes possible to obtain some detailed results about the structure of the dual spaces.