MATHEMATICAL MODAL LOGIC: A VIEW OF ITS EVOLUTION

60 Robert GoldblattAnother way to describe this conclusion is to say that if Alg(Λ) is the variety(equational class) of all modal algebras satisfying Λ, then in general CmS ∈ Alg(Λ)implies CmExS ∈ Alg(Λ). But CmExS = EmCmS, so the conclusion says thatAlg(Λ) contains the canonical embedding algebras of all its full complex algebras.This can then be strengthened, by applying duality theory, to show that Alg(Λ)contains the algebra EmA for any of its members A [Goldblatt, 1989, Theorem3.5.5]. Actually, to conclude that Alg(Λ) is closed under canonical embeddingalgebras it is enough to know that Λ is valid in the canonical frame S Λ κ for allinfinite cardinals κ. This follows by duality from the fact that S Λ κ is isomorphic tothe canonical structure CstA κ , where A κ is the free algebra in Alg(Λ) on κ-manygenerators, together with the fact that each member of Alg(Λ) is a homomorphicimage of some such free algebra.Ultimately this analysis can be generalised to any kind of Boolean algebra withoperators, to yield the following result:if C is any class of relational structures of the same type that is closedunder ultraproducts, then the variety of BAO’s generated by the classof algebras CmC is closed under canonical embedding algebras.This theorem was first formulated in [Goldblatt, 1989, Theorem 3.6.7], with aproof that used the important result of [Jónsson, 1967] on subdirectly irreduciblealgebras in congruence-distributive varieties and an obscure diagonal constructionon ultraproducts. An entirely different argument was given in [Goldblatt, 1991b]and analysed further in [Goldblatt, 1995]. It used the fact (2) from the previoussection, i.e. ExC ⊆ HPwC, and another formula,CstHSPCmC ⊆ SHUdPuC,which shows how the canonical structures of algebras from the variety generatedby CmC can themselves be built from members of C. When C is closed underultraproducts, so that PuC = C, this takes the formA ∈ HSPCmC implies CstA ∈ SHUdC,showing how canonical structures mediate between the dual operations on algebrasand structures. This result in turn depends on another fundamental fact,PuUbC ⊆ UbPuC,which states that the ultraproduct operation commutes with bounded unions. Astructure S is the bounded union of a collection {S j : j ∈ J} if the S j ’s are allinner substructures (subframes) of S and their union is S itself. This notion isdual to that of subdirect product, and indeed in the situation just described thereis a subdirect product representationCmS ↣ ∏ J CmS j