Sahlqvist's theorem for Boolean algebras with operators with an application to cylindric algebras

1. I n t r o d u c t i o n The aim of this note is to explain how a well-known result from Moda~ Logic, S~hlqvist's Theorem, can be applied in the theory of Boolean Algebras with Operators to obtain a large class of identities, called Sahlqvist identities, that are preserved under canonical embedding algebras. These identities can be specified as follows. Let a = { fi : i E I } be a set of (normal) additive operations. Let an untied term over cr be a term that is either (i) negative (i.e., in which every variable occurs in the scope of an odd number of complementat ion signs only), or (ii) of the form gl(g2.. . (gn(x)). . .) , where the g+s are+duals of unary elements of a (i.e., gi is defined by gi(x) = f i ( x ) for some unary operator in or), or Presented by I s t v ~ n N ~ m e t i ; Received December 15, 1993; Revised June 24, 1994 Studla Logica 54: 61-78, 1995. © 1995 KluwerAcademic Publishers. Printed in the Netherlands. 62 M. de Rijke, Y. Venema (iii) dosed (i.e., without occurrences of variables; note tha t this case is covered by (i)), or (iv) obtained from terms of type (i), (ii) or (iii) by applying + , . and eleraents of a only. Then, an equality is called a Sahlqvist equality if it is of the form s = 1, where s is obtained from complemented untied terms u by applying duals of elements of a to terms tha t have no variables in common, and • only. Before proceeding, let us give some examples and non-examples of Sahlqvist identit ies in algebraic logic. First of a~, the axioms governing normal , additive Boolean Algebras with Operators { fl : i e I } ( f i (x + y) = f ix + f i Y and fi0 = 0) are Sahlqvist identities. This should be obvious for the later axiom, while the former is equivalent to f i (x + y) . ( f i x + f i y ) <~ 0 and (f ix + f i y ) . f i ( x + y) <<. 0,