Algebra Universalis Full does not imply strong , does it ?

We give a duality for the variety of bounded distributive lattices that is not full (and therefore not strong) although it is full but not strong at the finite level. While this does not give a complete solution to the “Full vs Strong” Problem, which dates back to the beginnings of natural duality theory in 1980, it does solve it at the finite level. One consequence of this result is that although there is a Duality Compactness Theorem, which says that if an alter ego of finite type yields a duality at the finite level then it yields a duality, there cannot be a corresponding Full Duality Compactness Theorem. Is every full duality strong? This question is as old as the theory of natural dualities, and remains one of the most tantalising open problems in the foundations of the theory. A full duality is a special kind of dual equivalence between a quasi-variety A := ISP(M) of algebras and a category X := IScP+(M∼ ) of structured topological spaces. The earliest version of the “Full versus Strong” problem was formulated by Davey and Werner [13]: Question 1. If M∼ yields a full duality between A := ISP(M) and X := IScP +(M∼ ) does it follow that M∼ is injective in X? This question stems from a fundamental asymmetry in all known full dualities. For every full duality between A := ISP(M) and X := IScP+(M∼ ), the injectivity of the algebra M in A implies the injectivity of the alter ego M∼ in X. But the converse statement is false. Nevertheless, M∼ is injective in X in every known example of a full duality. (The interconnections between the injectivity of M and the injectivity of M∼ were discussed at length in [13]: see Proposition 1.11 on page 128 and pages 258–263 in the Appendix. The setting there was a general category-theoretic one and not specific to natural dualities. Some refinements in the setting of natural dualities are given in Exercises 6.2–6.6 of Clark and Davey [2].) The monograph [2] is recommended as the best source of basic facts as well as recent developments in the theory of natural dualities. Clark and Krauss [4] introduced the important notions of term-closed subsets of M and hom-closed subsets of M and proved that they are the same. They Presented by R. W. Quackenbush. Received October 1, 2002; accepted in final form November 10, 2004. 2000 Mathematics Subject Classification: 06D50; 08C05, 08C15, 18A40, 08A55.