On Some Compatible Operations on Heyting Algebras

Abstract

We study some operations that may be defined using the minimum operator in the context of a Heyting algebra. Our motivation comes from the fact that 1) already known compatible operations, such as the successor by Kuznetsov, the minimum dense by Smetanich and the operation G by Gabbay may be defined in this way, though almost never explicitly noted in the literature; 2) defining operations in this way is equivalent, from a logical point of view, to two clauses, one corresponding to an introduction rule and the other to an elimination rule, thus providing a manageable way to deal with these operations. Our main result is negative: all operations that arise turn out to be Heyting terms or the mentioned already known operations or operations interdefinable with them. However, it should be noted that some of the operations that arise may exist even if the known operations do not. We also study the extension of Priestley duality to Heyting algebras enriched with the new operations.