Duality for Boolean equalities

Stone duality, topological algebra, and recognition

For any Boolean space X and a discrete almost distributive lattice D, it is proved that the set C(X, D) of all continuous mappings of X into D, when D is equipped with the discrete topology, is an almost Boolean algebra under pointwise operations. Conversely, it is proved that any almost Boolean algebra is a homomorphic image of C(X,D) for a suitable Boolean space X and a discrete almost distributive lattice D.

Let π : Y → X \pi : Y\rightarrow X be a continuous surjection between compact Hausdorff spaces Y Y and X X which is irreducible in the sense that if F ⊊ Y F\subsetneq Y is closed, then π ( F ) ≠ X \pi (F)\neq X . We exhibit isomorphisms between various Boolean algebras associated to this data: the regular open sets of X X , the regular open sets of Y Y , the regular ideals of C ( X ) C(X) and the regular ideals of C ( Y ) C(Y) . We call X X and Y Y Boolean equivalent if the regular open sets of X X and the regular open sets of Y Y are isomorphic Boolean algebras. We give a characterization of when two compact metrizable spaces are Boolean equivalent; this characterization may be viewed as a topological version of the characterization of standard Borel spaces.

A monadic (Boolean) algebra is a Boolean algebra A together with an operator 3 on A (called an existential quantifier, or, simply, a quantifier) such that 30=0, pfk 3p, and 3(^A 3q) = 3p* 3g whenever p and q are in A. Most of this note uses nothing more profound about monadic algebras than the definition. The reader interested in the motivation for and the basic facts in the theory of monadic algebras may, however, wish to consult [2]. Every Boolean algebra can be converted into a monadic algebra, usually in several ways. (One way is to write 3p = p for all p; another is to write Bp =0 or 1 according p =0 or py^O. These special operators are known the discrete and the simple quantifier, respectively.) It follows, a fortiori, that every Boolean algebra can be embedded into a monadic algebra, and it is clear, on grounds of universal algebra, that among the monadic extensions of a Boolean algebra there is one that is as free possible. To be more precise, let us say that a monadic algebra A is a free monadic extension of a Boolean algebra B if (i) B is a Boolean subalgebra of A, (ii) A is (monadically) generated by B, (iii) every Boolean homomorphism g that maps B into an arbitrary monadic algebra C has a (necessarily unique) extension to a monadic homomorphism f that maps A into C. The statement that a monadic extension as free possible always exists means that every Boolean algebra B has a free monadic extension A; the algebra A is uniquely determined to within a monadic isomorphism that is equal to the identity on B. The purpose of what follows is to give a constructive proof of this fact, i.e., a proof that exhibits A by means of certain set-theoretic constructions based on B (instead of a structurally not very informative proof via equivalence classes of strings of symbols). A by-product of the proof is a rather satisfying insight into the structure of the (Stone) dual space of the free monadic extension. The theorem can (and will) be formulated in such a way to subsume the results of Hyman Bass [l ] on the cardinal number of finitely generated free monadic algebras. The idea of the construction is this. Step 1: form a free Boolean algebra generated by a copy of B. Step 2: adjoin the elements of that

The main purpose of this work is to study some infinite combinatorics applications in spaces of continuous functions, defined in Hausdorff compact spaces. Using infinite combinatorics in Boolean algebras, through Stone duality, we obtain a compact Hausdorff space. With certain properties in Boolean algebras it is possible to analyze analytic properties in the space of continuous functions defined in such space. Specifically, we analyze the Grothendieck property.

A classic result in the theory of Boolean algebras states that the dual space of the Boolean algebra of all subsets of a set U is homeomorphic to the Stone-Čech compactification of the discrete space U. It is shown in this chapter that the natural extension of this result to Boolean algebras with operators is false. In more detail, the analogue for Boolean algebras with operators of the Boolean algebra of subsets of a set U is the complex algebra \(\mathfrak{C}\mathfrak{m}(U)\) of a relational structure \(\mathfrak{U}\). It is shown that the dual space of \(\mathfrak{C}\mathfrak{m}(U)\) is not, in general, homeo-isomorphic to the Stone-Čech compactification of the structure \(\mathfrak{U}\) endowed with the discrete topology. However, a characterization of the dual space of the complex algebra \(\mathfrak{C}\mathfrak{m}(U)\) is given that may be viewed as a generalization of the Boolean result: the dual space of \(\mathfrak{C}\mathfrak{m}(U)\) is homeo-isomorphic to the Stone-Čech weak compactification of the discrete space \(\mathfrak{U}\). In fact, there is a lattice isomorphism between the lattice of equivalence classes of weak compactifications of the discrete space \(\mathfrak{U}\) and the lattice of subalgebras of the complex algebra \(\mathfrak{C}\mathfrak{m}(U)\) that contain all singleton subsets; and this lattice isomorphism maps the Stone-Čech weak compactification of \(\mathfrak{U}\) to the complex algebra \(\mathfrak{C}\mathfrak{m}(U)\) itself. If a relational structure is of finitary character, then the preceding result holds with the term “weak compactification” replaced by “compactification”. In particular, for Boolean algebras without operators we obtain the new result that there is an isomorphism from the lattice of equivalence classes of compactifications of a discrete space U to the lattice of subalgebras of the Boolean algebra of all subsets of U that contain the finite subsets of U. The arguments needed to establish these results are based upon a hybrid duality theory for homomorphisms that combines aspects of the algebraic and topological duality theories developed in Chapters 1 and 2 respectively.KeywordsWeak CompactificationFinite CharacterComplex AlgebraDiscrete TopologySubalgebrasThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

We deal with problems associated with Scott ranks of Boolean algebras. The Scott rank can be treated as some measure of complexity of an algebraic system. Our aim is to propound and justify the procedure which, given any countable Boolean algebra, will allow us to construct a Boolean algebra of a small Scott rank that has the same natural algebraic complexity as has the initial algebra. In particular, we show that the Scott rank does not always serve as a good measure of complexity for the class of Boolean algebras. We also study into the question as to whether or not a Boolean algebra of a big Scott rank can be decomposed into direct summands with intermediate ranks. Examples are furnished in which Boolean algebras have an arbitrarily big Scott rank such that direct summands in them either have a same rank or a fixed small one, and summands of intermediate ranks are altogether missing. This series of examples indicates, in particular, that there may be no nontrivial mutual evaluations for the Scott and Frechet ranks on a class of countable Boolean algebras.

A standard result in topological dynamics is the existence of minimal subsystem. It is a direct consequence of Zorn's lemma: given a compact topological space X with a map f: X→X, the set of compact non empty subspaces K of X such that f(K) ⊆ K ordered by inclusion is inductive, and hence has minimal elements. It is natural to ask for a point-free (or formal) formulation of this statement. In a previous work [3], we gave such a formulation for a quite special instance of this statement, which is used in proving a purely combinatorial theorem (van de Waerden's theorem on arithmetical progression).In this paper, we extend our analysis to the case where X is a boolean space, that is compact totally disconnected. In such a case, we give a point-free formulation of the existence of a minimal subspace for any continuous map f: X→X. We show that such minimal subspaces can be described as points of a suitable formal topology, and the “existence” of such points become the problem of the consistency of the theory describing a generic point of this space. We show the consistency of this theory by building effectively and algebraically a topological model. As an application, we get a new, purely algebraic proof, of the minimal property of [3]. We show then in detail how this property can be used to give a proof of (a special case of) van der Waerden's theorem on arithmetical progression, that is “similar in structure” to the topological proof [6, 8], but which uses a simple algebraic remark (Proposition 1) instead of Zorn's lemma. A last section tries to place this work in a wider context, as a reformulation of Hilbert's method of introduction/elimination of ideal elements.

Let X, Y be compact Hausdorff spaces and B£(X), B*(Y)f 0 < a, |3 < 12 (the first uncountable ordinal), the associated Banach spaces of bounded real-valued Baire functions of classes a and 0. If B*(X) =£ B*AX) (which is the case if a =£ 0 and X is not dispersed), then B*(X) is neither linearly isometric to BQ(Y) nor equivalent to B^(Y) in several other ways. B^(X) is linearly isometric to B%(Y) if and only if X is Baire isomorphic to Y. For 1 < a < ft the maximal ideal space of B£(X) for a nondispersed compact space X As not an F-space. 1. Let I bea compact (more generally, completely regular) Hausdorff space and C(X) the space of continuous real-valued functions on X. Let B0(X) = C(X)9 and inductively define Ba(X) for each ordinal a < Q, (Q denotes the first uncountable ordinal) to be the space of pointwise limits of sequences of functions in U | < a B^(X). Let B*(X) be the space of bounded functions contained in Ba(X). With the pointwise operations Ba(X) and B*(X) are lattice-ordered algebras. With the supremum norm B*(X) is a Banach algebra (see [4, §41]). The Baire sets of X of multiplicative class a, denoted by Za(X)9 are defined to be the zero sets of functions in B*(X). Those of additive class a, denoted by CZa(X), are defined as the complements of sets in Za(X). Finally, those of ambiguous class a, denoted by Aa(X), are the sets which are simultaneously in Za(X) and CZa(X). With the set-theoretic operations of union and intersection, Aa(X) is a Boolean algebra for each a < 12. The sets of exactly ambiguous class a, denoted by EAa(X are those in ^ a ( Z ) U ^ < a Ac(X). The sets of exactly additive and exactly multiplicative class a are defined analogously. The class of all Baire subsets of X is AMS (MOS) subject classifications (1970). Primary 06A65, 26A21, 28A05, 46E1S, 46E30, 46J10, 54C50, 54H05; Secondary 04A15.

On the existence of continuous (approximate) roots of algebraic equations

ON THE CONNECTION OF CYLINDRICAL HOMOMORPHISMS AND POINT FUNCTIONS FOR Crsα 's.

A strict implication calculus for compact Hausdorff spaces

The notion of a Boolean algebra with operators (BAO) was first defined by Jonsson and Tarski in 1951. Since that time, many varieties of BAOs, including modal algebras, closure algebras, monadic algebras, and of course relation algebras, have been studied. With the exception of relation algebras, these varieties each have one unary operator. This paper investigates a variety of BAOs with one binary operator. Begin with a semigroup S = $(S,\cdot).$ The complex algebra of S, denoted S$\sp+,$ is a Boolean algebra whose underlying set is the power set of S with set union, intersection and complementation as the Boolean algebra operations. The multiplication operation defined on the semigroup induces a normal, associative binary operation, '*', on the complex algebra as follows: for all subsets A and B contained in S, $A\* B=\{a\cdot b:a\in A, b\in B\}.$ Hence, the complex algebra of a semigroup is a BAO with one normal binary associative operator. Let S$\sp+$ be the class of all complex algebras of semigroups and consider the variety generated by the class S$\sp+,$ denoted ${\cal V}$(S$\sp+).$ The tools required to study this variety are developed, including the duality between BAOs and relational structures as it applies to ${\cal V}$(S$\sp+).$ A closure operator is defined which is used to determine homomorphic images of members of ${\cal V}$(S$\sp+).$ Theorems on the subdirectly irreducible and simples algebras in ${\cal V}$(S$\sp+)$ are proved. Next, the structure of ${\cal V}$(S$\sp+)$ is analyzed. The general problem of representing a BAO with one binary, normal, associative operator as a member of ${\cal V}$(S$\sp+)$ is discussed. Several examples and theorems concerning representation are presented. It is shown that the quasivariety generated by S$\sp+$ is strictly contained in ${\cal V}$(S$\sp+).$ Lastly, the structure of the lattice of subvarieties of ${\cal V}$(S$\sp+)$ is investigated. There are precisely two atoms in this lattice; each atom is generated by a two element algebra. Two infinite chains of varieties exist, with the smallest element in each chain a cover of exactly one atom. This leads to a discussion of certain splitting algebras and conjugate varieties. Finally, equations characterizing some important subvarieties are developed.

The theory of Boolean contact algebras has been used to represent a region based theory of space. Some of the primitives of Boolean algebras are not well motivated in that context. One possible generalization is to drop the notion of complement, thereby weakening the algebraic structure from Boolean algebra to distributive lattice. The main goal of this paper is to investigate the representation theory of that weaker notion, i.e., whether it is still possible to represent each abstract algebra by a substructure of the regular closed sets of a suitable topological space with the standard Whiteheadean contact relation.

It is well known that the laws of logic governing the sentence connectives—“and”, “or”, “not”, etc.—can be expressed by means of equations in the theory of Boolean algebras. The task of providing a similar algebraic setting for the full first-order predicate logic is the primary concern of algebraic logicians. The best-known efforts in this direction are the polyadic algebras of Halmos (cf. [2]) and the cylindric algebras of Tarski (cf. [3]), both of which may be described as Boolean algebras with infinitely many additional operations. In particular, there is a primitive operator, cκ, corresponding to each quantification, ∃υκ. In this paper we explore a version of algebraic logic conceived by A. H. Copeland, Sr., and described in [1], which has this advantage: All operators are generated from a finite set of primitive operations.Following the theory of cylindric algebras, we introduce, in the natural way, the classes of Copeland set algebras (SCpA), representable Copeland algebras (RCpA), and Copeland algebras of formulas. Playing a central role in the discussion is the set, Γ, of all equations holding in every set algebra. The reason for this is that the operations in a set algebra reflect the notion of satisfaction of a formula in a model, and hence an equation expresses the fact that two formulas are satisfied by the same sequences of objects in the model. Thus to say that an equation holds in every set algebra is to assert that a certain pair of formulas are logically equivalent.