Definitions for heterogeneous congruences and heterogeneous ideals on a Boolean module M are given and the respective lattices CongM and IdeM are presented. A characterization of the simple bijective Boolean modules is achieved differing from that given by Brink in a homogeneous approach. We construct the smallest and the greatest modular congruence having the same Boolean part. The same is established for modular ideals. The notions of kernel of a modular congruence and the congruence induced by a modular ideal are introduced to describe an isomorphism between CongM and IdeM. This isomorphism leads us to conclude that the class of the Boolean module is ideal determined.
The lattice CongD of all dynamic congruences on a given dynamic algebra D is presented. Whenever D is separable with zero we define dynamic ideal on D, given rise to the lattice IdeD. The notions of kernel of a dynamic congruence and the congruence generated by a dynamic ideal are introduced to describe a Galois connection between CongD and IdeD. We study conditions under which a dynamic congruence is determined by its kernel.
Remainders of subspaces are important e.g. in the realm of compactifications. Their extension to pointfree topology faces a difficulty: sublocale lattices are more complicated than their topological counterparts (complete atomic Boolean algebras). Nevertheless, the co-Heyting structure of sublocale lattices is enough to provide a counterpart to subspace remainders: the sublocale supplements. In this paper we give an account of their fundamental properties, emphasizing their similarities and differences with classical remainders, and provide several examples and applications to illustrate their scope. In particular, we study their behaviour under image and preimage maps, as well as their preservation by pointfree continuous maps (i.e. localic maps). We then use them to characterize nearly realcompact and nearly pseudocompact frames. In addition, we introduce and study hyper-real localic maps.
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. *
A characterization of the subdirectly irreducible separable dynamic algebras is presented. The notions developed for this study were also suitable to describe the previously found class of the simple separable dynamic algebras.
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