The Algebraic Face of Minimality

this paper we shall abstract from specific applications and investigate the behaviour of minimality operators from an algebraic perspective. Switching from the intended semantics for a language (e.g., models for first order logics, Kripke frames for modal logics) to algebras is a well understood and rather useful move, see e.g. Henkin et al. [11], Blok [1], and Goldblatt [7]. It enables us to use techniques from universal algebra to solve problems formulated in terms of the underlying logic. There is, however, at least one more motivation to investigate algebras induced by minimality operators: non--monotonic logic as well as logics involving conditionals are known to behave quite differently from standard logics like classical propositional logic, intuitionistic logic, or modal logics. This difference should be reflected (and at least partially explained) by means of the algebraic properties of the algebras induced by the minimality operator. For example, members of varieties (equationally definable classes) of algebras related to logics mostly have rather well behaved congruences (e.g. equationally definable principal congruences or at least first order definable congruences, see e.g. Blok and Pigozzi [2] and [3]). Properties of the congruences often reflect interesting properties of the associated logic. If it is true that there is an essential difference between non--monotonic and monotonic logics, then we should expect congruences to show some unusual features which do not appear in, say, Boolean algebras, Heyting algebras, or Modal algebras