Completions of B<scp>OOLEAN</scp> Algebras with operators

Monk, J. Donald

doi:10.1002/mana.19700460105

Cited by 42 publications

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“…Our development parallels that of Jónsson [1994]; we also need some lemmas from Monk [1970] concerning complete extensions of operators. Jónsson's approach can actually be given an axiomatic formulation so that it applies not only to canonical extensions and completions, but possibly also to other kinds of extensions that have not yet been considered.…”

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confidence: 99%

“…Monk [1970] extended the notion of the completion of a Boolean algebra to Boolean algebras with operators. Under the assumption that the operators of such an algebra A are completely additive, he showed that the completion of A always exists and is unique up to isomorphisms over A.…”

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confidence: 99%

“…The task of extending the theory of completions from Boolean algebras to Boolean algebras with operators was taken up by Monk [1970]. He showed that every Boolean algebra with operators A in which the operators are completely additive in each coordinate (that is, additive with respect to infinite joins -if they exist -as well as finite, non-empty joins) has a completion B (an expansion of the completion of the Boolean part of A) that is unique up to isomorphisms over A.…”

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The preservation of Sahlqvist equations in completions of Boolean algebras with operators

Givant

Venema

1999

Algebra Universalis

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Abstract. Monk [1970] extended the notion of the completion of a Boolean algebra to Boolean algebras with operators. Under the assumption that the operators of such an algebra A are completely additive, he showed that the completion of A always exists and is unique up to isomorphisms over A. Moreover, strictly positive equations are preserved under completions: a strictly positive equation that holds in A must hold in the completion of A.In this paper we extend Monk's preservation theorem by proving that certain kinds of Sahlqvist equations (as well as some other types of equations and implications) are preserved under completions. An example is given which shows that arbitrary Sahlqvist equations need not be preserved.In the study of Boolean algebras, it is often useful to pass from a given Boolean algebra A to an extension of A that is complete (in the sense that every set of elements, whether finite or infinite, has a supremum -a least upper boundand an infimum -a greatest lower bound -in A). Two rather different complete extensions of A are known from the literature. The first is the canonical (or perfect) extension: the Boolean algebra B of subsets of the collection of all ultrafilters of A. The second is the (MacNeille or Dedekind ) completion: the minimal Boolean algebra C that extends A and is complete. Each of these has its advantages and disadvantages. The advantage of the canonical extension B is that it is atomic. The disadvantage is that all proper infinite joins which do exist in A are "broken" in B; more precisely, if a is the supremum in A of an infinite set X, but not of any finite subset of X, then a cannot be the supremum of X in B. (This property is sometimes called compactness.) The advantage of the completion C is that all joins which do exist in A are preserved in C: if a is the supremum in A of any (finite or infinite) set X, then a is the supremum of X in C. The disadvantage is that C cannot be atomic unless A itself is atomic; in fact, the only atoms in C are the atoms of A.Jónsson and Tarski [1951] extended the theory of canonical extensions to Boolean algebras with additional operations that are additive in each coordinate, so-called Boolean algebras with operators. They showed that every Boolean algebra with operators A has a canonical extension B (an expansion of the canonical extension of the Boolean part of A) that is unique up to isomorphisms over A. Moreover, they also proved that every strictly positive equation -that is, every equation in which the complementation symbol does not occur -which holds in A must hold in B; in technical language, strictly positive equations are preserved under the passage to the canonical extension. (They also noted that some implications are similarly preserved.) Sahlqvist [1975] extended the Jónsson-Tarski preservation theorem to

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The preservation of Sahlqvist equations in completions of Boolean algebras with operators

Givant

Venema

1999

Algebra Universalis

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“…But a completion of a distributive lattice may not be a distributive lattice (e.g., [1]). In [13] Monk extended the definition of a completion to BAOs. Nowadays this construction is called the Dedekind-MacNeille completion, MacNeille completion or Monk completion.…”

Section: Introductionmentioning

confidence: 99%

“…Monk [13] proved that every positive equation is preserved under completions. Givant and Venema [8] extended this result by showing that all Sahlqvist equations are preserved under completions of conjugated BAOs (see also [18]).…”

Section: Introductionmentioning

confidence: 99%