Complete Atomic Boolean Algebra Research Articles

Profiniteness, monadicity and universal models in modal logic

Taking inspiration from the monadicity of complete atomic Boolean algebras, we prove that profinite modal algebras are monadic over Set. While analyzing the monadic functor, we recover the universal model construction - a construction widely used in the modal logic literature for describing finitely generated free modal algebras and the essentially finite generated subframes of their canonical models.

Difference–restriction algebras of partial functions with operators: Discrete duality and completion

We exhibit an adjunction between a category of abstract algebras of partial functions and a category of set quotients. The algebras are those atomic algebras representable as a collection of partial functions closed under relative complement and domain restriction; the morphisms are the complete homomorphisms. This generalises the discrete adjunction between the atomic Boolean algebras and the category of sets. We define the compatible completion of a representable algebra, and show that the monad induced by our adjunction yields the compatible completion of any atomic representable algebra. As a corollary, the adjunction restricts to a duality on the compatibly complete atomic representable algebras, generalising the discrete duality between complete atomic Boolean algebras and sets. We then extend these adjunction, duality, and completion results to representable algebras equipped with arbitrary additional completely additive and compatibility preserving operators.

On Stalnaker’s Simple Theory of Propositions

Robert Stalnaker recently proposed a simple theory of propositions using the notion of a set of propositions being consistent, and conjectured that this theory is equivalent to the claim that propositions form a complete atomic Boolean algebra. This paper clarifies and confirms this conjecture. Stalnaker also noted that some of the principles of his theory may be given up, depending on the intended notion of proposition. This paper therefore also investigates weakened constraints on consistency and the corresponding classes of Boolean algebras.

Remainders in pointfree topology

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 behavior 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.

Incomplete information and concept lattices

We present a mathematical model of sets with incomplete information about identity of elements and the membership relation. The model is based on L-valued sets where L is a complete atomic Boolean algebra. We present general foundations of the theory with emphasis on partially ordered sets and complete lattices. As an application, we show a method for constructing structures that represent the concept lattice of an incomplete binary dataset.

Double Approximation and Complete Lattices

We explore lattice theoretic aspects in rough set theory in terms of the duality between algebra and representation. Approximation spaces are dual to complete atomic Boolean algebras in the sense that there is an adjunction between corresponding suitable categories. This is an analogy with the adjunction between the category of topological spaces and the opposite of the category of frames in pointless topology. In this paper we consider a generalization of approximation spaces called double approximation systems. A double approximation system consists of a set and two equivalence relations on it. We construct an adjunction generalizing this concept for approximation spaces. To achieve this goal, we first introduce a natural generalization of complete atomic Boolean algebras called complete prime lattices. Then we select double approximation systems that can be dual to complete prime lattices and prove the adjunction.

N-ary Quantifiers and the Expressive Power of DP-Compositions

This paper, extending Keenan’s (1987, 1988) Case-extension of generalized quantifiers, proposes a natural algebraic semantics of DP-coordination and DP-composition. DPs in subject and non-subject positions are uniformly identified as a case-extension of a usual generalized quantifier, and DPs with different semantic cases combine with each other to yield polyadic quantifiers. The paper proves that each set of case-extensions forms a complete atomic Boolean algebra consisting of a full set of unary quantifiers, and it further shows that natural language requires the full power of binary quantification, i.e., the full set of (Fregean) reducible and unreducible binary quantifiers. This result, Type Effability, is derived from the fact that the set of all binary quantifiers can be constructed by taking the meet/join closure of the set of (composed) reducible type quantifiers. This fact is illustrated with non-constituent coordination constructions (e.g., gapping in English and Korean), whose interpretation requires arbitrary meets and joins of reducible binary quantifiers.

Algebraic Topological Closure Operators

For any closure operator c there is a T o -closure operator whose lattice of closed subsets are isomorphic to that of c. A correspondence between algebraic topological (T o ) closure operators on a nonempty set X and pre-orderes (partial orders) on X is established. Equivalent conditions are obtained for a T o -lattice to be a complete atomic Boolean algebra and for the lattice of closed subsets of an algebraic topological closure operator to be a complete atomic Boolean algebra. Further it is proved that a complete lattice is an algebraic T o -lattice if and only if it is isomorphic to the lattice of closed subsets of some algebraic topological closure operator on a suitable set.

Orthogonal sums of semigroups

The purpose of this paper is to prove that every semigroup with the zero is an orthogonal sum of orthogonal indecomposable semigroups. We prove that the set of all 0-consistent ideals of an arbitrary semigroup with the zero forms a complete atomic Boolean algebra whose atoms are summands in the greatest orthogonal decomposition of this semigroup.

How to construct extensional combinatory algebras

We develop a slight modification of Engeler's graph algebras, yielding extensional combinatory algebras. It is shown that by this construction we get precisely the class of Scott's D ∞-models generated by complete atomic Boolean algebras. In section 3 we construct extensional substructures of graph-algebras and Pω-models.

On the order scale of a uniform space

AbstractThe order topology is compact and T2 in both the scale and retracted scale of any uniform space (S, U). if (S, U) is T2 and totally bounded, the Samuel compactification associated with (S, U) can be obtained by uniformly embedding (S, U) in its order retracted scale (that is, the retracted scale with its order topology). This implies that every compact T2 space is both a closed subspace of a complete, infinitely distributive lattice in its order topology, and also a continuous, closed image of a closed subspace of a complete atomic Boolean algebra in its order topology.

The Duality of Distributive Continuous Lattices

Various aspects of the prime spectrum of a distributive continuous lattice have been discussed extensively in Hofmann-Lawson [7]. This note presents a perhaps optimally direct and self-contained proof of one of the central results in [7] (Theorem 9.6), the duality between distributive continuous lattices and locally compact sober spaces, and then shows how the familiar dualities of complete atomic Boolean algebras and bounded distributive lattices derive from it, as well as a new duality for all continuous lattices. As a biproduct, we also obtain a characterization of the topologies of compact Hausdorff spaces.Our approach, somewhat differently from [7], takes the open prime filters rather than the prime elements as the points of the dual space. This appears to have conceptual advantages since filters enter the discussion naturally, besides being a well-established tool in many similar situations.

Categories of frames for modal logic

§1. A complete atomic modal algebra (CAMA) is a complete atomic Boolean algebra with an additional completely additive unary operator. A (Kripke) frame is just a binary relation on a nonempty set. If is a frame, then is a CAMA, where mX = {y ∣ (∃x)(y < x Є X)}; and if is a CAMA then is a frame, where is the set of atoms of and b1 < b2 ⇔ b1 ∩ mb2 ≠∅.Now , and the validity of a modal formula on is equivalent to the satisfaction of a modal algebra polynomial identity by and conversely, so the validity-preserving constructions on frames ought to be in some sense equivalent to the identity-preserving constructions on CAMA's. The former are important for modal logic, and many of the results of universal algebra apply to the latter, so it is worthwhile to fix precisely the sense of the equivalence.The most important identity-preserving constructions on CAMA's can be described in terms of homomorphisms and complete homomorphisms. Let and be the categories of CAMA's with homomorphisms and complete homomorphisms, respectively. We shall define categories and of frames with appropriate morphisms, and show them to be dual respectively to and . Then we shall consider certain identity-preserving constructions on CAMA's and attempt to describe the corresponding validity-preserving constructions on frames.The proofs of duality involve some rather detailed calculations, which have been omitted. All the category theory a reader needs to know is in the first twenty pages of [7].

An algebraic characterization of power set in countable standard models of ZF

The following is a classical result:Theorem 1.1. A complete atomic Boolean algebra is isomorphic to a power set algebra [2, p. 70].One of the consequences of [3] is: If M is a countable standard model of ZF and is a countable (in M) model of a complete ℵ0-categorical theory T, then there is a countable standard model N of ZF and a Λ ∈ N such that the Boolean algebra of definable (in T with parameters from ) subsets of is isomorphic to the power set algebra of Λ in N. In particular if and T the theory of equality with additional axioms asserting the existence of at least n distinct elements for each n < ω, then there is an N and Λ ∈ N with 〈PN(Λ), ⊆〉 isomorphic to the countable, atomic, incomplete Boolean algebra of the finite and cofinite subsets of ω.From the above we see that some incomplete Boolean algebras can be realized as power sets in standard models of ZF.Definition 1.1. A countable Boolean algebra 〈B, ≤〉 is a pseudo-power set if there is a countable standard model of ZF, N and a set Λ ∈ N such thatIt is clear that a pseudo-power set is atomic.

On the Lattice of Primitive Convergence Structures

Let S be any set and denote by F(S) the collection of all fiters on S. The collection A(S) of all mappings from F(S) to 2s, 2s being ordered by the dual of its usual ordering, may be regarded as a product of complete Boolean algebras and is, therefore, a complete atomic Boolean algebra [4]. A(S) is called the lattice of primitive convergence structures on S. If q ∈ A(S) and , then is said to q-converge to a point x ∈ S if . The collection of all topologies on S may be identified with a subset of A(S); this subset of A(S) will be denoted by T(S). A more specialized class of primitive convergence structures, and one which properly contains T(S), is C(S), the subcomplete lattice of all convergence structures on S.