Priestley Configurations and Heyting Varieties

We investigate Heyting varieties determined by prohibition of systems of configuraations in Priestley duals; we characterize the configuration systems yielding such varieties. On the other hand, the question whether a given finitely generated Heyting variety is obtainable by such means is solved for the special case of systems of trees. Priestley duality provides a correspondence between bounded distributive lattices and certain ordered topological spaces. (See [11] and [12], and for an elementary exposition [4].) In a previous article [2] we showed, among other things, that the class of all Heyting algebras whose Priestley spaces contain no copy of a given single (finite) configuration formed a variety iff the configuration was a tree. Consequently, prohibiting any system of trees presents a Heyting variety as well. But if a system of prohibited configurations has more than one element, it need not consist of trees alone, or be replaceable by such, to yield a variety. One of our two main results characterizes just such configuration systems. Our other major result addresses the opposite question: given a variety, when can it be obtained by prohibiting a system of trees? (The general question of when a variety results from the prohibition of any system of configurations is beyond the scope of this article.) We present a complete characterization for the case of finitely generated varieties; special consideration is given to varieties generated by a single object. There are two topics that we needed to discuss in some detail, in our context for more or less technical purposes, but these subjects may be of some interest in their own right. One of them is the nature of the Priestley equivalents of Heyting homomorphisms, and the other is The second author would like to express his thanks for the support by the project LN 00A056 of the Ministry of Education of the Czech Republic, by the NSERC of Canada and by the Gudder Trust of the University of Denver. The third author would like to express his thanks for the support by the NSERC of Canada and a partial support by the project LN 00A056 of the Ministry of Education of the Czech Republic. 1 2 RICHARD N. BALL, ALEŠ PULTR, AND JIŘÍ SICHLER the order structure of coproducts of Priestley spaces, a topic which in general seems to be far from fully understood. To each of these we devote a special section.