An algebraic approach to canonical formulas: modal case
The authors generalize the result that the category MA of modal algebras and modal homomorphisms is dually equivalent to the category MS of modal spaces and continuous \(p\)-morphisms to the following result: Theorem 3.4: MA\(^R\) is dually equivalent to MS\(^P\). Here MA\(^R\)is the category of modal algebras and relativized modal algebra homomorphisms, and MS\(^P\) is the category of modal spaces and partial continuous \(p\)-morphisms. For a modal algebra \((A, \wedge, \vee, \neg, \diamond, 0,1)\) and \(s\in A\), \(A_s\) is defined by \(A_s = [0,s]=\{ x\in A\,|\,0\leq x\leq s\}\), and the operations on \(A_s\) are defined as follows:{\parindent=8mm \begin{itemize}\item[1)] \(x\wedge_s y = x\wedge y\) \item[2)] \(x\vee_s y = x\vee y\) \item[3)] \(0_s = 0\) \text{and} \ \(1_s = s\) \item[4)] \(\neg_s x = \neg x \wedge s\) \item[5)] \(\diamond_s x= s \wedge \diamond x\) \end{itemize}} \(A_s\) is called a relativization of \(A\) to \(s\). For modal algebras \(A,B\) and a map \(\eta : A\to B\), \(\eta\) is called a relativized modal algebra homomorphism if it is a modal algebra homomorphism from \(A\) to the relativized modal algebra \(B_{\eta(1)}\). For Stone spaces \(X\) and \(Y\), a partial map \(f:X\to Y\) is called a partial continuous map if dom(\(f\)) is a clopen subset of \(X\) and \(f\) is a continuous map from dom(\(f\)) to \(Y\). Let \((X,R)\) and \((Y,Q)\) be modal spaces and \(f:X\to Y\) be a partial map. Then \(f\) is called a partial continuous \(p\)-morphism if: 1. \(x,z\in \text{dom}(f)\) and \(xRz\) imply \(f(x)Qf(z)\); 2. \(x\in \text{dom}(f)\) and \(f(x)Qy\) imply that there exists \(z\in \text{dom}(f)\) such that \(xRz\) and \(f(z)=y\). Moreover, by considering MA as the transitive case K4 of modal algebra, they give a new algebraic proof of Zakharyashev's theorem that each logic over K4 is axiomatizable by canonical frames.
scientific article; zbMATH DE number 5977619
scientific article