Relationship of Algebraic Theories to Powerset Theories and Fuzzy Topological Theories for Lattice-Valued Mathematics

This paper deals with a broad question—to what extent is topologyalgebraic—using two specific questions: (1) what are the algebraicconditions on the underlying membership lattices which insure thatcategories for topology and fuzzy topology are indeed topologicalcategories; and (2) what are the algebraic conditions which insure thatalgebraic theories in the sense of Manes are a foundation for thepowerset theories generating topological categories for topology and fuzzytopology? This paper answers the first question by generalizing the Höhle-Šostak foundations for fixed-basis lattice-valued topology and the Rodabaugh foundations for variable-basis lattice-valued topology using semi-quantales; and it answers the second question by givingnecessary and sufficient conditions under which certain theories—the veryones generating powerset theories generating (fuzzy) topological theories inthe sense of this paper—are algebraic theories, and these conditions useunital quantales. The algebraic conditions answering the second question aremuch stronger than those answering the first question. The syntacticbenefits of having an algebraic theory as a foundation for the powersettheory underlying a (fuzzy) topological theory are explored; therelationship between these two specific questions is discussed; the role ofpseudo-adjoints is identified in variable-basis powerset theories which arealgebraically generated; the relationships between topological theories inthe sense of Adámek-Herrlich-Strecker and topological theories in the sense of this paper are fully resolved; lower-image operators introduced for fixed-basismathematics are completely described in terms of standard image operators;certain algebraic theories are given which determine powerset theoriesdetermining a new class of variable-basis categories for topology and fuzzytopology using new preimage operators; and the theories of this paper areundergirded throughout by several extensive inventories of examples