Algebra univers. 59 (2008) 31–47 0002-5240/08/010031 – 17, published online October 11, 2008 DOI 10.1007/s00012-008-2035-7 c Birkh¨ auser Verlag, Basel, 2008 Algebra Universalis Priestley configurations and Heyting varieties Richard N. Ball, Aleˇ s Pultr, and Jiˇ r´ ı Sichler Dedicated to George Gr¨ atzer and E. Tam´ as Schmidt on their 70th birthdays Abstract. We investigate Heyting varieties determined by prohibition of systems of con- figurations in Priestley duals; we characterize the configuration systems yielding such vari- eties. 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. 1. Introduction Priestley duality ([11], [12], for an elementary exposition see [5]) provides a cor- respondence between bounded distributive lattices and certain ordered topological spaces. In a previous article [2] we showed, a.o., that the class of all Heyting alge- bras whose Priestley spaces contain no copy of a given (finite) configuration formed a variety if and only if 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 does not necessarily have to consist of trees only (or be replaceable by such) to yield a variety. One of the two principal aims of this article is to characterize such configuration systems. In the second one we address the opposite question: given a variety, when can it be obtained by prohibiting a system of trees? (The general question of whether it 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. Presented by B. Davey. Received October 23, 2003; accepted in final form October 24, 2006. 2000 Mathematics Subject Classification : Primary: 06D20, 06A10; Secondary: 54F05. Key words and phrases : distributive lattice, Priestley duality, poset, Heyting algebra, variety. The second author would like to express his thanks for the support by the project 1M0021620808 of the Ministry of Education of the Czech Republic, by the NSERC of Canada and by the University of Denver. The third author would like to express his thanks for the support by the NSERC of Canada and partial support by the project 1M0021620808 of the Ministry of Education of the Czech Republic. 31