Complete Boolean algebra

Last updated

In mathematics, a complete Boolean algebra is a Boolean algebra in which every subset has a supremum (least upper bound). Complete Boolean algebras are used to construct Boolean-valued models of set theory in the theory of forcing. Every Boolean algebra A has an essentially unique completion, which is a complete Boolean algebra containing A such that every element is the supremum of some subset of A. As a partially ordered set, this completion of A is the Dedekind–MacNeille completion.

Contents

More generally, if κ is a cardinal then a Boolean algebra is called κ-complete if every subset of cardinality less than κ has a supremum.

Examples

Complete Boolean algebras

Non-complete Boolean algebras

Now let a0, a1, … be pairwise disjoint infinite sets of naturals, and let A0, A1, … be their corresponding equivalence classes in P(ω)/Fin. Then given any upper bound X of A0, A1, … in P(ω)/Fin, we can find a lesser upper bound, by removing from a representative for X one element of each an. Therefore the An have no supremum.

Properties of complete Boolean algebras

The completion of a Boolean algebra

The completion of a Boolean algebra can be defined in several equivalent ways:

The completion of a Boolean algebra A can be constructed in several ways:

If A is a metric space and B its completion then any isometry from A to a complete metric space C can be extended to a unique isometry from B to C. The analogous statement for complete Boolean algebras is not true: a homomorphism from a Boolean algebra A to a complete Boolean algebra C cannot necessarily be extended to a (supremum preserving) homomorphism of complete Boolean algebras from the completion B of A to C. (By Sikorski's extension theorem it can be extended to a homomorphism of Boolean algebras from B to C, but this will not in general be a homomorphism of complete Boolean algebras; in other words, it need not preserve suprema.)

Free κ-complete Boolean algebras

Unless the Axiom of Choice is relaxed, [1] free complete boolean algebras generated by a set do not exist (unless the set is finite). More precisely, for any cardinal κ, there is a complete Boolean algebra of cardinality 2κ greater than κ that is generated as a complete Boolean algebra by a countable subset; for example the Boolean algebra of regular open sets in the product space κω, where κ has the discrete topology. A countable generating set consists of all sets am,n for m, n integers, consisting of the elements x  κω such that x(m) < x(n). (This boolean algebra is called a collapsing algebra, because forcing with it collapses the cardinal κ onto ω.)

In particular the forgetful functor from complete Boolean algebras to sets has no left adjoint, even though it is continuous and the category of Boolean algebras is small-complete. This shows that the "solution set condition" in Freyd's adjoint functor theorem is necessary.

Given a set X, one can form the free Boolean algebra A generated by this set and then take its completion B. However B is not a "free" complete Boolean algebra generated by X (unless X is finite or AC is omitted), because a function from X to a free Boolean algebra C cannot in general be extended to a (supremum-preserving) morphism of Boolean algebras from B to C.

On the other hand, for any fixed cardinal κ, there is a free (or universal) κ-complete Boolean algebra generated by any given set.

See also

Related Research Articles

<span class="mw-page-title-main">Axiom of choice</span> Axiom of set theory

In mathematics, the axiom of choice, abbreviated AC or AoC, is an axiom of set theory equivalent to the statement that a Cartesian product of a collection of non-empty sets is non-empty. Informally put, the axiom of choice says that given any collection of sets, each containing at least one element, it is possible to construct a new set by choosing one element from each set, even if the collection is infinite. Formally, it states that for every indexed family of nonempty sets, there exists an indexed set such that for every . The axiom of choice was formulated in 1904 by Ernst Zermelo in order to formalize his proof of the well-ordering theorem.

<span class="mw-page-title-main">Boolean algebra (structure)</span> Algebraic structure modeling logical operations

In abstract algebra, a Boolean algebra or Boolean lattice is a complemented distributive lattice. This type of algebraic structure captures essential properties of both set operations and logic operations. A Boolean algebra can be seen as a generalization of a power set algebra or a field of sets, or its elements can be viewed as generalized truth values. It is also a special case of a De Morgan algebra and a Kleene algebra.

<span class="mw-page-title-main">Cardinal number</span> Size of a possibly infinite set

In mathematics, a cardinal number, or cardinal for short, is what is commonly called the number of elements of a set. In the case of a finite set, its cardinal number, or cardinality is therefore a natural number. For dealing with the case of infinite sets, the infinite cardinal numbers have been introduced, which are often denoted with the Hebrew letter (aleph) marked with subscript indicating their rank among the infinite cardinals.

In mathematical logic, model theory is the study of the relationship between formal theories, and their models. The aspects investigated include the number and size of models of a theory, the relationship of different models to each other, and their interaction with the formal language itself. In particular, model theorists also investigate the sets that can be defined in a model of a theory, and the relationship of such definable sets to each other. As a separate discipline, model theory goes back to Alfred Tarski, who first used the term "Theory of Models" in publication in 1954. Since the 1970s, the subject has been shaped decisively by Saharon Shelah's stability theory.

<span class="mw-page-title-main">Power set</span> Mathematical set containing all subsets of a given set

In mathematics, the power set (or powerset) of a set S is the set of all subsets of S, including the empty set and S itself. In axiomatic set theory (as developed, for example, in the ZFC axioms), the existence of the power set of any set is postulated by the axiom of power set. The powerset of S is variously denoted as P(S), 𝒫(S), P(S), , or 2S. Any subset of P(S) is called a family of sets over S.

In mathematics, pointless topology, also called point-free topology and locale theory, is an approach to topology that avoids mentioning points, and in which the lattices of open sets are the primitive notions. In this approach it becomes possible to construct topologically interesting spaces from purely algebraic data.

In mathematics, a topological space is called separable if it contains a countable, dense subset; that is, there exists a sequence of elements of the space such that every nonempty open subset of the space contains at least one element of the sequence.

This is a glossary of some terms used in the branch of mathematics known as topology. Although there is no absolute distinction between different areas of topology, the focus here is on general topology. The following definitions are also fundamental to algebraic topology, differential topology and geometric topology. For a list of terms specific to algebraic topology, see Glossary of algebraic topology.

<span class="mw-page-title-main">Complete lattice</span> Partially ordered set in which all subsets have both a supremum and infimum

In mathematics, a complete lattice is a partially ordered set in which all subsets have both a supremum (join) and an infimum (meet). A conditionally complete lattice satisfies at least one of these properties for bounded subsets. For comparison, in a general lattice, only pairs of elements need to have a supremum and an infimum. Every non-empty finite lattice is complete, but infinite lattices may be incomplete.

In mathematics, a cofinite subset of a set is a subset whose complement in is a finite set. In other words, contains all but finitely many elements of If the complement is not finite, but is countable, then one says the set is cocountable.

In the mathematical area of order theory, the compact elements or finite elements of a partially ordered set are those elements that cannot be subsumed by a supremum of any non-empty directed set that does not already contain members above the compact element. This notion of compactness simultaneously generalizes the notions of finite sets in set theory, compact sets in topology, and finitely generated modules in algebra.

In abstract algebra, an interior algebra is a certain type of algebraic structure that encodes the idea of the topological interior of a set. Interior algebras are to topology and the modal logic S4 what Boolean algebras are to set theory and ordinary propositional logic. Interior algebras form a variety of modal algebras.

In mathematics, a field of sets is a mathematical structure consisting of a pair consisting of a set and a family of subsets of called an algebra over that contains the empty set as an element, and is closed under the operations of taking complements in finite unions, and finite intersections.

In topology and related areas of mathematics, a topological property or topological invariant is a property of a topological space that is invariant under homeomorphisms. Alternatively, a topological property is a proper class of topological spaces which is closed under homeomorphisms. That is, a property of spaces is a topological property if whenever a space X possesses that property every space homeomorphic to X possesses that property. Informally, a topological property is a property of the space that can be expressed using open sets.

In the mathematical field of set theory, Martin's axiom, introduced by Donald A. Martin and Robert M. Solovay, is a statement that is independent of the usual axioms of ZFC set theory. It is implied by the continuum hypothesis, but it is consistent with ZFC and the negation of the continuum hypothesis. Informally, it says that all cardinals less than the cardinality of the continuum, 𝔠, behave roughly like ℵ0. The intuition behind this can be understood by studying the proof of the Rasiowa–Sikorski lemma. It is a principle that is used to control certain forcing arguments.

In mathematical logic, a Boolean-valued model is a generalization of the ordinary Tarskian notion of structure from model theory. In a Boolean-valued model, the truth values of propositions are not limited to "true" and "false", but instead take values in some fixed complete Boolean algebra.

In mathematics, a free Boolean algebra is a Boolean algebra with a distinguished set of elements, called generators, such that:

  1. Each element of the Boolean algebra can be expressed as a finite combination of generators, using the Boolean operations, and
  2. The generators are as independent as possible, in the sense that there are no relationships among them that do not hold in every Boolean algebra no matter which elements are chosen.

In model theory and related areas of mathematics, a type is an object that describes how a element or finite collection of elements in a mathematical structure might behave. More precisely, it is a set of first-order formulas in a language L with free variables x1, x2,..., xn that are true of a set of n-tuples of an L-structure . Depending on the context, types can be complete or partial and they may use a fixed set of constants, A, from the structure . The question of which types represent actual elements of leads to the ideas of saturated models and omitting types.

This is a glossary of terms and definitions related to the topic of set theory.

References

  1. Stavi 1974.

Literature