When is a natural duality `good'?
Both authors, jointly as well as with various collaborators, have been working in the area of dualities for some time. Recently they developed a powerful machinery to construct so-called ``strong dualities'' between finitely generated quasivarieties and certain categories of compact topological algebras [J. Aust. Math. Soc., Ser. A 58, 248-280 (1995; Zbl 0839.08005)]. If \(M\) is the underlying set of an algebra generating a quasivariety \(\mathcal A\), a judicious, but not necessarily unique, choice of total operations, partial operations and relations on \(M\) together with the discrete topology turns \(M\) into a topological algebra of a certain type that via non-empty products, compact substructures and isomorphic copies generates a category \(\mathcal X\) dually equivalent to \(\mathcal A\). However, even a ``simple'' \(\mathcal A\)-structure on \(M\) can lead to a rather ``complicated'' topological algebra structure. The present paper, after a brief introduction to natural and strong dualities, investigates properties of the \(\mathcal A\)-structure on \(M\) that will result in a relatively ``simple'' topological algebra structure. The results have been condensed into ten theorems, which are not intended to be exhaustive. ``Simple'' here usually means that the number of total operations, or partial operations, or relations needed is either 0 or at most 1. E.g., we can dispense with partial operations iff the generator of \(\mathcal A\) is injective. If, in addition, epis are surjective in \(\mathcal A\), we can eliminate the relations as well. One exception is the so-called Logarithmic Duality Theorem, where besides the numbers of total and partial operations the type of relations is restricted to those that ``avoid products''. In this case a nice description of coproducts in \(\mathcal X\) is possible.
scientific article; zbMATH DE number 883422
scientific article