Copeland algebras

Neat embedding theorems yield an abstract algebraic characterization for the representability of a given class of algebras by set algebras. Resek and Thompson’s theorem called attention to a new kind of representation in the theory of cylindric algebras, to the representation by cylindric relativised set algebras. In this paper, we present the algebraic characterization of this kind of representation; we formulate neat embedding theorems for this representation.

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.

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.

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 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.

The algebras studied in this paper were suggested to the author by William Craig as a possible substitute for cylindric algebras. Both kinds of algebras may be considered as algebraic versions of first-order logic. Cylindric algebras can be introduced as follows. Let ℒ be a first-order language, and let be an ℒ-structure. We assume that ℒ has a simple infinite sequence ν0, ν1, … of individual variables, and we take as known what it means for a sequence ν0, ν1, … of individual variables, and we take as known what it means for a sequence x = 〈x0, x1, …〉 of elements of to satisfy a formula ϕ of ℒ in . Let ϕ be the collection of all sequences x which satisfy ϕ in . We can perform certain natural operations on the sets ϕ, of basic model-theoretic significance: Boolean operations = cylindrifications diagonal elements (0-ary operations) . In this way we make the class of all sets ϕ into an algebra; a natural abstraction gives the class of all cylindric set algebras (of dimension ω). Thus this method of constructing an algebraic counterpart of first-order logic is based upon the notion of satisfaction of a formula by an infinite sequence of elements. Since, however, a formula has only finitely many variables occurring in it, it may seem more natural to consider satisfaction by a finite sequence of elements; then ϕ becomes a collection of finite sequences of varying ranks (cf. Tarski [10]). In forming an algebra of sets of finite sequences it turns out to be possible to get by with only finitely many operations instead of the infinitely many ci's and dij's of cylindric algebras. Let be the class of all algebras of sets of finite sequences (an exact definition is given in §1).

Color algebras and affine connections on S6

We extend the theory of profinite algebras, as it is used in the algebraic theory of regular languages, to the more general setting of Stone topological algebras with the ultimate goal of going beyond classes of regular languages. We introduce Stone pseudovarieties, that is, classes of Stone topological algebras of a fixed topological signature that are closed under taking Stone quotients, closed subalgebras and finite direct products. Looking at the dual Stone spaces of Boolean algebras of subsets of a given topological algebra, we find a simple characterization of when the dual space admits a natural structure of topological algebra; the characterization is given in terms of how the Boolean algebra behaves under the inverses of the operation evaluation mappings of each arity. This provides an alternative, which is presented in the form of an equivalence of categories, to the duality theory of M. Gehrke. As an application, a Stone quotient of a Stone topological algebra that is residually in a given Stone pseudovariety is shown to be also residually in it, thereby extending the corresponding result of M. Gehrke for the Stone pseudovariety of all finite algebras over discrete signatures, which was recently extended to topological signatures jointly by the authors and H. Goulet-Ouellet. The residual closure of a Stone pseudovariety is thus a Stone pseudovariety, and these are precisely the Stone analogues of varieties. A Birkhoff type theorem for Stone varieties is also established and it is shown how Reiterman’s theorem can be derived from it.

The notion of polyadic algebra was introduced by Halmos to reflect algebraically the predicate logic without equality. Later Halmos enriched the study with the introduction of the notion of equality. These algebras are very closely related to the cylindric algebras of Tarski. The notion of diagonal free cylindric algebra predates that of cylindric algebra and is also due to Tarski. The theory of diagonal free algebras forms an important fragment of the theories of polyadic and cylindric algebras.

There is no algorithm for determining whether or not an equation is true in every 3-dimensional cylindric algebra. This theorem completes the solution to the problem of finding those values of α and β for which the equational theories of CAα and RCAβ are undecidable. (CAα and RCAβ are the classes of α-dimensional cylindric algebras and representable β-dimensional cylindric algebras. See [4] for definitions.) This problem was considered in [3]. It was known that RCA0 = CA0 and RCA1 = CA1 and that the equational theories of these classes are decidable. Tarski had shown that the equational theory of relation algebras is undecidable and, by utilizing connections between relation algebras and cylindric algebras, had also shown that the equational theories of CAα and RCAβ are undecidable whenever 4 ≤ α and 3 ≤ β. (Tarski's argument also applies to some varieties K ⊆ RCAβ with 3 ≤ β and to any variety K such that RCAα ⊆ K ⊆ CAα and 4 ≤ α.)Thus the only cases left open in 1961 were CA2, RCA2 and CA3. Shortly there-after Henkin proved, in one of Tarski's seminars at Berkeley, that the equational theory of CA2 is decidable, and Scott proved that the set of valid sentences in a first-order language with only two variables is recursive [11]. (For a more model-theoretic proof of Scott's theorem see [9].) Scott's result is equivalent to the decidability of the equational theory of RCA2, so the only case left open was CA3.

Methods and results from the representation theory of finite di- mensional algebras have led to many interactions with other areas of mathe- matics. Such areas include the theory of Lie algebras and quantum groups, commutative algebra, algebraic geometry and topology, and in particular the new theory of cluster algebras. The aim of this workshop was to further de- velop such interactions and to stimulate progress in the representation theory of algebras.

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.

A Galois correspondence between cylindric set algebras and permutation groups is presented in this paper.Moreover, the Galois connection is used to help establish two important algebraic properties for certain classes of finite-dimensional cylindric algebras, namely the amalgamation property and the property that epimorphisms are surjective.The importance of the amalgamation property (AP for short) in algebraic logic has been recognized for a long time.In [16] the connection with the interpolation property of first-order logic is discussed.The positive amalgamation results from the author's thesis [2] and their extensions announced in [3] are cited in 2.4 of [16].These results are established, with a slight improvement, in 4 below.The key to this study is a lattice anti-isomorphism between subgroups of the symmetric group 5 nd subalgebras of the full set algebra 31 (a, p) when 0</i<a + l<.This Galois theory is developed in 2.It is applied in 3 to show that the algebra 9(o, p), 0</i<a + l<w, is homogeneous and the variety it generates has enough injectives.The announcement [3] dealt with the case ju < a.In this treatment of Galois theory we also include the results of H. Andrka and I. Nmeti which show that the theory works for p = a + 1, but for no larger p.In 5 the property (ES) that all epimorphisms are surjective is established for certain varieties of CAa's.It has been shown by I. Nmeti (see [14]) that the ES property for classes of CAa's is related to Beth's definability result.The question of when the ES property holds was raised in [8, p. 311, Problem 10].To put the results in this paper in perspective, perhaps an additional remark will be useful.Many of the results assert that some property (that depends on a and p.) holds if /i < a + 1 < (o.Upon reading a preliminary draft of the paper, Andrka and Nmeti showed that most of the main results could not be improved in the sense that examples show that the properties fail for a > p. > a + 1.In particular, this applies to Theorems 2.2, 2.8, 2.9, 3.7, 3.8(2), Corollaries 3.10, 5.5 and Lemma 5.2(2).Essentially the only result where the value of p is not known to be the best possible is in Theorem 2.5.

Complexity of Boolean algebras

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