Concurrent algebras: an algebraic study of a fragment of concurrent propositional dynamic logic

摘要

In this paper we will study the algebraic semantics of a simplified fragment of the Concurrent Propositional Dynamic Logic CPDL introduced by R. Goldblatt. We shall define a concurrent algebra as a Boolean algebra A endowed with two unary operators □ and Δ such that (1) -〈A,□〉 is a normal modal algebra, (2) 〈A,Δ〉 is a monotonic modal algebra, and (3) □(a → b) ≤Δa → Δb, and □0 ∨ Δ1 = 1, for every a, b ∈ A. We shall prove that every concurrent algebra is isomorphic to a subalgebra of a full complex algebra 〈P(X),□_R,Δ_R〉, where X is a set, and R is a subset of X × P(X). In others words, we get that the variety CA of concurrent algebras is generated by the class of all full complex algebras. Also we introduce the Kripke neighbourhood monotonic frames, or kn-frames. We will prove that this class is interdefinable with the class of concurrent frames. We shall prove that there exists a topological duality between concurrent algebras and concurrent spaces. We will also introduce descriptive kn-frames, and we will prove that these structures are interdefinable with concurrent spaces. Using the representation theory, we shall prove that the variety CA has the Amalgamation Property.