The preservation of Sahlqvist equations in completions of Boolean algebras with operators

Abstract

Abstract. Monk [1970] extended the notion of the completion of a Boolean algebra to Boolean algebras with operators. Under the assumption that the operators of such an algebra A are completely additive, he showed that the completion of A always exists and is unique up to isomorphisms over A. Moreover, strictly positive equations are preserved under completions: a strictly positive equation that holds in A must hold in the completion of A.In this paper we extend Monk's preservation theorem by proving that certain kinds of Sahlqvist equations (as well as some other types of equations and implications) are preserved under completions. An example is given which shows that arbitrary Sahlqvist equations need not be preserved.In the study of Boolean algebras, it is often useful to pass from a given Boolean algebra A to an extension of A that is complete (in the sense that every set of elements, whether finite or infinite, has a supremum -a least upper boundand an infimum -a greatest lower bound -in A). Two rather different complete extensions of A are known from the literature. The first is the canonical (or perfect) extension: the Boolean algebra B of subsets of the collection of all ultrafilters of A. The second is the (MacNeille or Dedekind ) completion: the minimal Boolean algebra C that extends A and is complete. Each of these has its advantages and disadvantages. The advantage of the canonical extension B is that it is atomic. The disadvantage is that all proper infinite joins which do exist in A are "broken" in B; more precisely, if a is the supremum in A of an infinite set X, but not of any finite subset of X, then a cannot be the supremum of X in B. (This property is sometimes called compactness.) The advantage of the completion C is that all joins which do exist in A are preserved in C: if a is the supremum in A of any (finite or infinite) set X, then a is the supremum of X in C. The disadvantage is that C cannot be atomic unless A itself is atomic; in fact, the only atoms in C are the atoms of A.Jónsson and Tarski [1951] extended the theory of canonical extensions to Boolean algebras with additional operations that are additive in each coordinate, so-called Boolean algebras with operators. They showed that every Boolean algebra with operators A has a canonical extension B (an expansion of the canonical extension of the Boolean part of A) that is unique up to isomorphisms over A. Moreover, they also proved that every strictly positive equation -that is, every equation in which the complementation symbol does not occur -which holds in A must hold in B; in technical language, strictly positive equations are preserved under the passage to the canonical extension. (They also noted that some implications are similarly preserved.) Sahlqvist [1975] extended the Jónsson-Tarski preservation theorem to

Preservation of Sahlqvist fixed point equations in completions of relativized fixed point Boolean algebras with operators

“…Thus, we obtain the result of [8] as a corollary of our main proof. Our proof seems to be different from the one of [8] and [18] and is similar to the proofs of [15], [2], and [4], whereas Venema and Givant's approach is closer to the approach of [10]. Our proof, however, is also different (and in a way simpler) than the proofs of [15] and [4], for a few reasons.…”

“…Monk [13] proved that every positive equation is preserved under completions. Givant and Venema [8] extended this result by showing that all Sahlqvist equations are preserved under completions of conjugated BAOs (see also [18]). Givant and Venema also gave an example of a Sahlqvist equation not preserved under non-conjugated BAOs.…”

“…We could define the upper extension of a map f on a BAO A by f (b) = A {f (c) : c ∈ A, c ≥ b}. As follows from [8] (see also [9, Finally, we note that the skeleton of a Sahlqvist fixed point term is an Lterm. Fixed point operators are only allowed in the negative terms substituted in the skeleton.…”

Sahlqvist Correspondence for Modal mu-calculus

“…These properties are of course related, and further algebraic properties of Sahlqvist formulas have been established (e.g., [15]). The celebrated 'Esakia lemma' [12] is used in a key step in the proof of completeness (e.g., [23]).…”

Extending Łukasiewicz Logics with a Modality: Algebraic Approach to Relational Semantics

“…The algebraic treatment of this canonicity result was considered in [23]. This success lead mathematicians to consider so called Sahlqvist equations in wider contexts (e.g., [17,19,9]). …”