Bulletin of the Section of Logic Volume 28/2 (1999), pp. 99–107 Ivo D¨ untsch Ewa Orlowska MIXING MODAL AND SUFFICIENCY OPERATORS ∗ Abstract We explore Boolean algebras with sufficiency operators, and investigate a class of mixed algebras which corresponds to frames < U, R > with a modal operator 〈R〉 and a sufficiency operator [[R]]. 1. Introduction It is well known that not every first order property can be expressed as a statement of modal logic, a case in point being irreflexivity. Noting that a relation is irreflexive if and only if its complement is reflexive – and reflexivity is modally expressible – Humberstone [1] introduced an “inaccessibility ” operator, which was determined by the complement of a frame relation; a similar idea was put forward by Gargov et. al. [2] who used a “sufficiency” operator. As they point out, having separate modal and sufficiency operators only mirrors the deficiency of one construction in the other, and thus, mixing operators will lead to higher expressivity. In this note, we will present a representation theorem for frames and algebras with a sufficiency operator along the lines of the corresponding theorem for Boolean algebras with operators [3], and exhibit the duality between the category of frames with suitable morphisms and the sufficiency algebras in parallel to the modal case [4]. We will define a class of mixed * The authors gratefully acknowledge support by the KBN/British Council Grant No WAR/992/151