Standard topological quasi-varieties

This study addresses a problem which lies at the confluence of algebra, topology and mathematical logic. It is motivated by the theory of natural dualities, which provides a tight connection between a quasi-variety generated by a finite algebra M and the topological quasi-variety generated by a related topological structure ∼ . We introduce the notion of a standard topological quasi-variety and initiate a program of study to determine which topological quasi-varieties are standard and which are not. We say that a topological quasi-variety is standard if, in an appropriate sense, there is a nice axiomatic description of its members. Knowing in advance that it is standard allows us to recognize its members by looking only at their finite substructures. Let M∼ = 〈M ; G, H, R, T 〉 be a finite structure with operations G, partial operations H, relations R and discrete topology T. The topological quasi-variety generated by M∼ is the category QT(M∼ ) := IScP + M∼ of isomorphic copies of topologically closed substructures of non-zero direct powers, with the product topology, of M∼ . Interest in topological quasi-varieties stems from the fact that they arise as the duals to algebraic quasi-varieties under natural dualities, and the name comes from their obvious structural similarity to algebraic quasi-varieties. (See Clark and Davey [4], Clark and Krauss [5].) A natural duality is a special kind of dual equivalence between the quasi-variety Q(M) := ISPM generated by a finite algebra M and a topological quasi-variety QT(M∼ ) = IScP + M∼ generated by a structure M∼ having the same underlying set as M. The general theory of natural dualities provides methods to produce, from the algebra M, a structure M∼ that will yield a natural duality on Q(M). 2000 Mathematics Subject Classification. 08C15, 08A60, 03B99, 06D50, 22A30.