On an unsolved problem from the theory of Boolean algebras

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.

Using ♢ ℵ 1 {\diamondsuit _{{\aleph _1}}} we construct a Boolean algebra B B of power ℵ 1 {\aleph _1} , with the following properties: (a) B B has just ℵ 1 {\aleph _1} subalgebras. (b) Every uncountable subset of B B contains a countable independent set, a chain of order type η \eta , and three distinct elements a , b a,b and c c , such that a ∩ b = c a \cap b = c . (a) refutes a conjecture of J. D. Monk, (b) answers a question of R. McKenzie. B B is embeddable in P ( ω ) P(\omega ) . A variant of the construction yields an almost Jónson Boolean algebra. We prove that every subalgebra of an interval algebra is retractive. This answers affirmatively a conjecture of B \text {B} . Rotman. Assuming MA \text {MA} or the existence of a Suslin tree we find a retractive BA \text {BA} not embeddable in an interval algebra. This refutes a conjecture of B. Rotman. We prove that an uncountable subalgebra of an interval algebra contains an uncountable chain or an uncountable antichain. Assuming CH \text {CH} we prove that the theory of Boolean algebras in Magidor’s and Malitz’s language is undecidable. This answers a question of M. Weese.

The isolation of dimensions from a data matrix has been traditionally formulated in terms of an al gebraic or geometric model. Order analysis was de veloped as a method of multidimensional analysis and scaling based on the theory of Boolean algebra. The order analytic algorithm utilizes functions of the propositional calculus in lieu of eigenvalues and eigenvectors of the general linear model. Also, the graphic presentation of latent space in coordinates of the Euclidian space is paralleled in ordering- theoretic models by dendrograms of the test space. A conceptual outline of order analysis is presented, followed by an empirical comparison of factor and order analysis solutions of a sample data problem. Resulting factor and order analytic structures are evaluated in terms of meeting criteria of simple structure and correct reflection of broad cognitive categories. In addition, the relations of proximity and dominance are discussed from the perspectives of both Cartesian and Leibnitzian theories of di mensionality as pertaining to problems of multi variate analysis and scaling.

By limiting the range of the predicate variables in a second-order language, one may obtain restricted versions of second-order logic such as weak second-order logic or definable subset logic. In this note, we provide an infinitary strongly complete axiomatization for several systems of this kind having the range of the predicate variables as a parameter. The completeness argument uses simple techniques from the theory of Boolean algebras. This article is dedicated to our friend John N. Crossley on the occasion of his 86th birthday.

Often, we assume that an action is permitted simply because it is not explicitly forbidden; or, similarly, that an action is forbidden simply because it is not explicitly permitted. This kind of assumptions appear, e.g., in autonomous computing systems where decisions must be taken in the presence of an incomplete set of norms regulating a particular scenario. Combining default and deontic reasoning over actions allows us to formally reason about such assumptions. With this in mind, we propose a logical formalism for default reasoning over a deontic action logic. The novelty of our approach is twofold. First, our formalism for default reasoning deals with actions and action operators, and it is based on the deontic action logic originally proposed by Segerberg. Second, inspired by Segerberg's approach, we use tools coming from the theory of Boolean Algebra. These tools allow us to extend Segerberg's algebraic completeness result to the setting of Default Logics.

By using the universality of the theory of irreflexive symmetric graphs the universality of the theory of Boolean algebras with the Hartig quantifier is shown. Let b be any Boolean algebra with infinitely many atoms. Then the theory of b with the Hartig quantifier is undecidable.

The research aims to analyze geodetic methods for determining the location of natural reserve areas and standardizing the obtained data. The object of the research is the standardization of geodetic data for determining the boundaries of natural reserve areas. Normative and legal documents related to the classification and determination of territories of natural reserve areas and the main geodetic methods used to determine the boundaries of such areas were analyzed in the article. The structure and description of the main geodetic methods for the practical implementation of work to determine the location of natural reserve areas, including geodetic measurements, global navigation, and remote sensing methods, were also discussed. In addition, a number of discrepancies were identified in the process of obtaining, storing, and reproducing geodetic data. Overall, the article contains important information about methods for determining the boundaries of natural reserve areas and how mathematical methods can be used to standardize geodetic data by formalizing it. The research showed that formalization of geodetic data can be done using mathematical methods such as mathematical logic, Boolean algebra, and category theory. The article proposes the use of mathematical methods of data formalization, such as mathematical logic, Boolean algebra, and category theory, to structure and standardize geodetic data. Analytical and graphical models for standardizing geodetic data for determining the boundaries of natural reserve areas were developed using mathematical methods of formalization such as mathematical logic, Boolean algebra, and category theory.

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 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.

The theory of discriminator algebras and varieties has been investigated extensively, and provides us with a wealth of information and techniques applicable to specific examples of such algebras and varieties. Here we give several such examples for Boolea

This is a paper on Stone duality in computer science with special focus on topics with applications in formal language theory. In Section 2 we give a general overview of Stone duality in its various forms: for Boolean algebras, distributive lattices, and frames. For distributive lattices, we discuss both Stone and Priestley duality. We identify how to move between the different dualities and which dual spaces carry the Scott topology. We then focus on three themes. The first theme is additional operations on distributive lattices and Boolean algebras. Additional operations arise in denotational semantics in the form of predicate transformers. In verification they occur in the form of modal operators. They play an essential role in Eilenberg's variety theorem in the form of quotient operations. Quotient operations are unary instantiations of residual operators which are dual to the operations in the profinite algebras of algebraic language theory. We discuss additional operations in Section 3. The second theme is that of hyperspaces, that is, spaces of subsets of an underlying space. Some classes of algebras may be seen as the class of algebras for a functor. In the case of predicate transformers the dual functors are hyperspace constructions such as the Plotkin, Smyth, and Hoare powerdomain constructions. The algebras-for-a-functor point of view is central to the coalgebraic study of modal logic and to the solution of domain In the algebraic theory of formal languages various hyperspace-related product constructions, such as block and Schutzenberger products, are used to study complexity hierarchies. We describe a construction, similar to the Schutzenberger product, which is dual to adding a layer of quantification to formulas describing formal languages. We discuss hyperspaces in Section 4. The final theme is that of equations. These are pairs of elements of dual spaces. They arise via the duality between subalgebras and quotient spaces and have provided one of the most successful tools for obtaining decidability results for classes of regular languages. The perspective provided by duality allows us to obtain a notion of equations for the study of arbitrary formal languages. Equations in language theory is the topic of Section 5.

We analyze Zorn's Lemma and some of its consequences for Boolean algebras in a constructive setting. We show that Zorn's Lemma is persistent in the sense that, if it holds in the underlying set theory, in a properly stated form it continues to hold in all intuitionistic type theories of a certain natural kind. (Observe that the axiom of choice cannot be persistent in this sense since it implies the law of excluded middle.) We also establish the persistence of some familiar results in the theory of (complete) Boolean algebras—notably, the proposition that every complete Boolean algebra is an absolute subretract. This (almost) resolves a question of Banaschewski and Bhutani as to whether the Sikorski extension theorem for Boolean algebras is persistent.

Complexity of Boolean algebras

Studying Boolean algebras with distinguished subalgebras, we establish the existence of continuum many simple superatomic Boolean algebras with distinguished subalgebra whose elementary theories are distinct and each of them lacks countably saturated models. The subalgebras coincide with the Boolean algebras modulo their Frechet ideals.

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.