A A topological characterization of an almost Boolean algebra

Introduction.Boolean equalities occur naturally in Logic and especially in the theory of polyadic algebras.For instance, they have been used in [2] under the name of Champ Logique.The purpose of this paper is to develop a duality theory for Boolean equalities and by doing so establish a representation theory for them.0. Notation.The basic notation for Boolean algebras is the standard one (for instance, see [l]).Throughout this paper, the letter O will denote the two-element Boolean algebra; whenever a topology is assumed to be defined on 0, then this topology is the discrete one.If 73 is a Boolean algebra, then a valuation of B is a homomorphism from 73 onto 0. We make precise the terminology we shall be using in connection with the Stone duality theory for Boolean algebras.A Boolean space is a compact Hausdorff space whose topology is generated by the clopen sets.If X is a Boolean space, then the dual algebra of X is the Boolean algebra of clopen sets of X.If 73 is a Boolean algebra, then the dual space of 73 is the Boolean space X of all valuations of 73 (the topology on X is the topology generated by all sets of the form {v: vp = 1} where p belongs to B).

The motivation for this paper comes from the following sources. First, one can observe that the two major concepts underlying the methods of reasoning with incomplete information are the concept of degree of truth of a piece of information and the concept of approximation of a set of information items. We shall refer to the theories employing the concept of degree of truth as to theories of fuzziness and to the theories employing the concept of approximation as to theories of roughness (see [6] for a survey). The algebraic structures relevant to these theories are residuated lattices ([7], [12], [13], [16], [17], [18], [22], [23]) and Boolean algebras with operators ([19], [21], [10], [11]), respectively. Residuated lattices provide an arithmetic of degrees of truth and Boolean algebras equipped with the appropriate operators provide a method of reasoning with approximately determined information. Both classes of algebras have a lattice structure as a basis. Second, both theories of fuzziness and theories of roughness develop generalizations of relation algebras to algebras of fuzzy relations [20] and algebras of rough relations ([4], [5], [9]), respectively. In both classes a lattice structure is a basis. Third, not necessarily distributive lattices with modal operators, which can be viewed as most elementary approximation operators, are recently developed in [24] (distributive lattices with operators are considered in [14] and [25]). With this background, our aim is to begin a systematic study of the classes of algebras that have the structure of a (not necessarily distributive) lattice and, moreover, in each class there are some operators added to the lattice which are relevant for binary relations. Our main interest is in developing relational representation theorems for the classes of lattices with operators under consideration. More precisely, we wish to guarantee that each algebra of our classes is isomorphic to an algebra of binary relations on a set. We prove the theorems of that form by suitably extending the Urquhart representation theorem for lattices ([26]) and the representation theorems presented in [1]. The classes defined in the paper are the parts which put together lead to what might be called lattice-based relation algebras. Our view is that these algebras would be the weakest structures relevant for binary relations. All the other algebras of binary relations considered in the literature would then be their signature and/or axiomatic extensions.Throughout the paper we use the same symbol for denoting an algebra or a relational system and their universes.KeywordsBinary RelationDistributive LatticeRepresentation TheoremResiduated LatticeRelation AlgebraThese 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.

It is well known that in a distributive lattice L, every filter is an intersection of prime filters, and dually. Moreover, L is relatively complemented if and only if every prime filter (ideal) is maximal. In 1947, A. Monteiro [3] posed the question of determining whether a distributive lattice, in which every filter is an intersection of maximal filters, and dually, is necessarily relatively complemented. This question also appears as Problem 39 in G. Gr/itzer [2; p. 156]. Monteiro states that J. Dieudonn6 and L. Nachbin have independently shown that neither the condition that every filter is an intersection of maximal filters, nor its dual, implies the other. We will solve this problem in the negative by constructing a non-Boolean distributive lattice L which satisfies the above conditions on the filters and ideals L. The referee was most helpful in simplifying the explanation of the construction of L. This lattice is necessarily infinite for if a finite distributive lattice is such that each of its filters is an intersection of maximal filters then it is a Boolean algebra. Indeed, this follows from the fact that if F, F1,..., F, are prime filters and F = Of="1 Fi then F=Fto for some io~ {1,..., n}.

Stone duality, topological algebra, and recognition

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.

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.

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.

Some aspects of the theory of Boolean algebras and distributive lattices–in particular, the Stone Representation Theorems and the properties of filters and ideals–are analyzed in a constructive setting.

For finite automata as coalgebras in a category C , we study languages they accept and varieties of such languages. This generalizes Eilenberg’s concept of a variety of languages, which corresponds to choosing as C the category of Boolean algebras. Eilenberg established a bijective correspondence between pseudovarieties of monoids and varieties of regular languages. In our generalization, we work with a pair C / D of locally finite varieties of algebras that are predual, i.e., dualize on the level of finite algebras, and we prove that pseudovarieties of D -monoids bijectively correspond to varieties of regular languages in C . As one instance, Eilenberg’s result is recovered by choosing D = sets and C = Boolean algebras. Another instance, Pin’s result on pseudovarieties of ordered monoids, is covered by taking D = posets and C = distributive lattices. By choosing as C amp;equals; D the self-predual category of join-semilattices, we obtain Polák’s result on pseudovarieties of idempotent semirings. Similarly, using the self-preduality of vector spaces over a finite field K , our result covers that of Reutenauer on pseudovarieties of K -algebras. Several new variants of Eilenberg’s theorem arise by taking other predualities, e.g., between the categories of non-unital Boolean rings and of pointed sets. In each of these cases, we also prove a local variant of the bijection, where a fixed alphabet is assumed and one considers local varieties of regular languages over that alphabet in the category C .

POLYNOMIAL NORMAL FORMS AND THE EMBEDDING OF POLYNOMIAL ALGEBRAS

REMARKS ON FULLY INVARIANT CONGRUENCES

A certain class of models is considered, i.e. sets with relations specifically adapted to solve a number of problems connected with various questions of imbeddings of partially ordered sets in Boolean algebras and distributive lattices. A special place in this class is occupied by a subclass called the class of Boolean models by the author.Bibliography: 15 titles.

This paper calculates, in a precise way, the complexity of the index sets for computable structures that are autostable relative to strong constructivizations and belong to one of the following classes: linear orderings, Boolean algebras, distributive lattices, partial orderings, rings, and commutative semigroups. We also calculate the complexity of the index set for computable structures that have computable dimension n, where n is a fixed natural number greater than 1.

Lattices can be defined either as special partially ordered sets or as algebras. In this chapter, we present both definitions and show their equivalence. We study several special classes of lattices: modular and distributive lattices and complete lattices. The last part of the chapter is dedicated to Boolean algebras and Boolean functions.

LetLbe a distributive lattice andMn,q(L)(Mn(L), resp.) the semigroup (semiring, resp.) ofn×q(n×n, resp.) matrices overL. In this paper, we show that if there is a subdirect embedding from distributive latticeLto the direct product∏i=1m‍Liof distributive latticesL1,L2, …,Lm, then there will be a corresponding subdirect embedding from the matrix semigroupMn,q(L)(semiringMn(L), resp.) to semigroup∏i=1m‍Mn,q(Li)(semiring∏i=1m‍Mn(Li), resp.). Further, it is proved that a matrix over a distributive lattice can be decomposed into the sum of matrices over some of its special subchains. This generalizes and extends the decomposition theorems of matrices over finite distributive lattices, chain semirings, fuzzy semirings, and so forth. Finally, as some applications, we present a method to calculate the indices and periods of the matrices over a distributive lattice and characterize the structures of idempotent and nilpotent matrices over it. We translate the characterizations of idempotent and nilpotent matrices over a distributive lattice into the corresponding ones of the binary Boolean cases, which also generalize the corresponding structures of idempotent and nilpotent matrices over general Boolean algebras, chain semirings, fuzzy semirings, and so forth.