Monadic dynamic algebras

Abstract

The main purpose of this work is to introduce the class of the monadic dynamic algebras (dynamic algebras with one quantifier). Similarly to a theorem of Kozen we establish that every separable monadic dynamic algebra is isomorphic to a monadic (possibly non-standard) Kripke structure. We also classify the simple (monadic) dynamic algebras. Moreover, in the dynamic duality theory, we analyze the conditions under which a hemimorphism of a dynamic algebra into itself defines a quantifier.Monadic Boolean algebras [1] and dynamic algebras [10,11] are both recognizable as modal algebras [2,3]. The fact that a monadic Boolean algebra is a very particular case of a dynamic algebra with only one action (the quantifier) led us to the study of a class of dynamic algebras, which we will call monadic dynamic algebras, where there exists an action behaving as a quantifier. Functional monadic Boolean algebrasFunctional monadic Boolean algebras were introduced by Halmos in [1]. The set B X of all functions from X (a non-empty set) to B, where B = (B, ∨, ∼, 0) is a Boolean algebra, is itself a Boolean algebra B X with respect to the pointwise operations, namely, if p and q are elements of B X , then the supremum p ∨ q and the complement ∼ p are defined byfor every x ∈ X, and where the zero and the unit of B X are, respectively, functions that are constantly equal to 0 and 1. Let us denote by R(p) the range of the function p of B X . A Boolean subalgebra A of B X such that (i) for every p in A the supremum R(p) and the infimum R(p) exist in B and (ii) the (constant) functions ∃p and ∀p, defined bybelong to A, is called a functional monadic Boolean algebra or a B-valued functional monadic Boolean algebra with domain X. In this definition it is not necessary to impose that, for every p in A, both ∃p and ∀p exist and belong to A, since ∀p can be interpreted as ∼ (∃(∼ p)) and ∃p as ∼ (∀(∼ p)). By the mutual duality of these operators we shall study ∃ alone. In particular, in the functional monadic Boolean algebra 2 X , where 2 = {0, 1}, if p is a non-zero element of 2 X , there exists an element x 0 in X such that p(x 0 ) = 1. Then 1 ∈ R(p) and ∃p(x) = R(p) = 1, for every x ∈ X. If p = 0 in 2 X , then p(x) = 0, for every x ∈ X, and ∃p(x) = R(p) = 0. Therefore 2 X is a functional monadic Boolean algebra, where ∃p = 1, for p = 0, and ∃0 = 0. *