New tools for proving dualizability

We address the problem of proving that a finite algebraM is dualizable, or strongly dualizable, in the sense of [1]. There are generally two aspects to the problem: (1) handling the finite members of SP(M), and (2) handling the infinite members of SP(M). In principle, the problem rests entirely at the infinite level (that is, in the second aspect). In practice, however, the real work is typically done at the finite level; then the analysis is “lifted ” to the infinitely level by a suitable combinatorial argument. In this paper we give two “lifting theorems, ” one for dualizability and the other for strong dualizability, which may prove useful in this enterprise. 1. Dualizability Throughout this paper M will denote a fixed finite algebra. The theory of natural dualities attempts to produce a “nice ” dual category for the quasivariety generated by M. See the book [1] for the full story. In this and the next section, we shall give a bare minimum of definitions needed to state our theorems. By an algebraic relation of M is meant any subuniverse of Mn for some n < ω, construed as an n-ary relation on the set M. Fix R, a set of algebraic relations of M. Definition 1.1. (1) M ∼ is the relational-topological structure 〈M,R, Td〉, whose universe is M (the universe ofM), whose relations are those in R, and whose topology Td is the discrete topology. (2) If A ∈ SP(M), then D(A) denotes the induced relational-topological sub-structure of M∼ A with universe Hom(A,M). (D(A) is the dual of A.) Fix A ∈ SP(M) and a ∈ A. The “evaluation-at-a ” map Hom(A,M)→M, given by h 7 → h(a), is an example of a continuous R-preserving map from D(A) to M ∼. Definition 1.2. (1) R dualizes M if for every A ∈ SP(M), the continuous R-preserving maps from D(A) to M ∼ are precisely the evaluation maps. (2) R dualizes M at the finite level if the previous item holds for every A