Interior algebras and topology - Naturman, Colin Ashley (Department of Mathematics and Applied Mathematics, 2016)
In this thesis connections between categories of interior algebras and categories of topological spaces, and generalizations of topological concepts to interior algebras, are investigated. The following are some of the most significant results we obtain: The establishment of a duality between topological spaces and complete atomic interior algebras formalized in terms of a category-theoretic co-equivalence between the category of topological spaces and continuous maps and the category of complete atomic interior algebras and maps known as complete topomorphisms (Theorem 2.1.7). Under this co-equivalence, continuous open maps correspond to complete homomorphisms (Theorem 2.1.8). We also establish a duality between arbitrary interior algebras and structures known as Stone fields in terms of a co-equivalence between the category of interior algebras and topomorphisms (see Definition 1.1.8) and the category of Stone fields and their morphisms the field maps (Theorem 2.2.14). Under this co-equivalence weakly open field maps (see Definition 2.2.17) correspond to homomorphisms (Theorem 2.2.18). The well-known connection between pre-ordered sets and interior algebras is shown to be a special case of topological duality (see section 4 of chapter 2). The topological concepts of neighbourhoods, convergence and accumulation are generalized to interior algebras (Chapter 3), and are used to generalize the topological separation and compactness properties to interior algebras (Chapter 4 and Chapter 5). What is particularly interesting with regard to the separation properties is that most of them are first order properties of interior algebras (see Theorem 4.5.11). This should be contrasted with the situation for frames/locales [12] and topological model theory [10]. By generalizing the concept of α-separation to interior algebras we obtain an ω chain of strictly elementary classes of interior algebras all of which have hereditarily undecidable first order theories (Theorem 4.3.14). Characterizations of irreducibility properties for interior algebras are also found. These properties (subdirect irreducibility, finite subdirect irreducibility, direct indecomposability, simplicity and semi-simplicity) can be characterized in many different ways. Characterizations in terms of open elements (fixed points of the interior operator) are found (Theorem 1.3.18 and Theorem 1.3.21) and these are used to obtain further characterizations. In particular a characterization in terms of topological properties of Stone spaces of interior algebras is obtained (Theorem 2.3.9). We also find characterizations of the irreducibility properties in the power set interior algebras of topological spaces (Theorem 2.1.15) and in interior algebras obtained from pre-ordered sets (Theorem 2.4.16). What is particularly striking is that the irreducibility properties correspond to very natural topological properties. (Other results characterizing or related to the irreducibility properties are 2.4.11, 2.4.17, 5.1.13, and 5.1.15). Bibliography: pages 134-135.