MATHEMATICAL MODAL LOGIC: A VIEW OF ITS EVOLUTION

Mathematical Modal Logic: A View of its Evolution 57a relational model as a structure (K, R, H), where H is a collection of truthvaluationsΦ on (K, R) in Kripke’s sense that satisfies certain closure properties.That did not produce a full equivalence between algebras and models. A languageindependent-approach was taken by Thomason [1972b] who defined a “first-ordersemantics” using structures S = (K, R, P ), where P is a collection of subsets of Kthat forms a subalgebra of the full complex algebra Cm(K, R). This subalgebra istaken in place of Cm(K, R) as the algebra assigned to S. Validity in S is definedas truth in all models M = (S, Φ) on S satisfying the constraint that the setM(p) = {x : Φ(p, x) = ⊤} belongs to P for all variables p.By imposing suitable restrictions on P , essentially set-theoretic versions of theconditions (i)–(iii) of section 3.3 that defined the Jónsson-Tarski perfect extensions,a notion of “descriptive” frame (K, R, P ) is arrived at. This theory was developedin [Goldblatt, 1974], where the descriptive frames were shown to form a categorydually equivalent to Malg. A topological approach to duality for closure algebrasand quasi-orderings was independently investigated by Leo Èsakia [1974].Connections between relational structures and algebras can be convenientlyexpressed in the “calculus” of class operations. We use the symbols S, H, andUd for the operations of closing a class of structures under subframes, p-morphicimages, and disjoint unions, respectively. Pu and Pw are used for closure underultraproducts and ultrapowers, whileCmC = {A : A ∼ = CmS for some S ∈ C}is the class of all (isomorphic copies of) complex algebras of structures in the classC. Then the isomorphism (1) above implies that CmUdC = PCmC for any class Cof frames. Similarly, the representation( ∏ J S j) /F −→ ( ∐ J S j) J /Ffrom section 6.3 of an ultraproduct of frames as a subframe of an ultrapower of adisjoint union yields the conclusion that in generalPuC ⊆ SPwUdC.There are numerous properties that can be express in this way using class operations,for exampleSHC ⊆ HSC, SCmHC = SCmC, SUdC = UdSC, PuSHC ⊆ HSPuC.An inventory of such facts may be found in [Goldblatt, 1995; Goldblatt, 2000].Dual to the formation of the algebra EmA = CmCstA is the association withany structure S of its canonical extension ExS = CstCmS, a structure whosepoints are the ultrafilters on the underlying set of S (hence ExS is sometimescalled the ultrafilter extension of S). There is a p-morphismS J /F ↠ ExS