Quasivarieties of Metric Algebras

Khudyakov, V. A.

doi:10.1023/b:allo.0000004176.74753.9e

Cited by 2 publications

(10 citation statements)

References 2 publications

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“…However the use of closed seems confusing since we also use it for the closedness under algebraic operations, therefore we prefer the word continuous. As we will see in Section 5, this terminology is consistent with continuous quasivariety defined in [16].…”

Section: Proposition 315 ([21]mentioning

confidence: 92%

“…In fact, the idea of using indexed binary relations to axiomatize metric structures is already in the literature of universal algebra [21,16] under the name of metric algebra. This notion is slightly wider than that of quantitative algebra in the sense that operations in metric algebras are not required to be non-expansive.…”

Section: Metric and Quantitative Algebramentioning

confidence: 99%

“…This notion is slightly wider than that of quantitative algebra in the sense that operations in metric algebras are not required to be non-expansive. Weaver [21] and Khudyakov [16] prove continuous versions of the characterization theorem for quasivarieties, i.e., classes of algebras defined by implications, and the decomposition theorem corresponding to the one in the classical theory.…”

Section: Metric and Quantitative Algebramentioning

confidence: 99%

“…Proposition 2. 16. Given an ultrafilter U on I and a family of real numbers (a i ) i∈I , then lim i→U a i exists, i.e., lim inf i→U a i and lim sup i→U a i coincide.…”

Section: Filter and Limitmentioning

confidence: 99%

“…Definition 3. 16. A class K of metric algebras is called continuous if it is closed under taking ultraproducts.…”

Section: Proposition 315 ([21]mentioning

confidence: 99%

See 4 more Smart Citations

Continuous Varieties of Metric and Quantitative Algebras

Hino

2017

Preprint

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A metric algebra is a metric variant of the notion of Σ-algebra, first introduced in universal algebra to deal with algebras equipped with metric structures such as normed vector spaces. In this paper, we showed metric versions of the variety theorem, which characterizes strict varieties (classes of metric algebras defined by metric equations) and continuous varieties (classes defined by a continuous family of basic quantitative inferences) by means of closure properties. To this aim, we introduce the notion of congruential pseudometric on a metric algebra, which corresponds to congruence in classical universal algebra, and we investigate its structure.

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Section: Proposition 315 ([21]mentioning

confidence: 92%

Section: Metric and Quantitative Algebramentioning

confidence: 99%

Section: Metric and Quantitative Algebramentioning

confidence: 99%

“…Proposition 2. 16. Given an ultrafilter U on I and a family of real numbers (a i ) i∈I , then lim i→U a i exists, i.e., lim inf i→U a i and lim sup i→U a i coincide.…”

Section: Filter and Limitmentioning

confidence: 99%

“…Definition 3. 16. A class K of metric algebras is called continuous if it is closed under taking ultraproducts.…”

Section: Proposition 315 ([21]mentioning

confidence: 99%

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Continuous Varieties of Metric and Quantitative Algebras

Hino

2017

Preprint

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Varieties of Metric and Quantitative Algebras

Hino

2016

Preprint

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Metric algebras are metric variants of Σ-algebras. They are first introduced in the field of universal algebra to deal with algebras equipped with metric structures such as normed vector spaces. Recently a similar notion of quantitative algebra is used in computer science to formulate computational effects of probabilistic programs. In this paper we show that varieties of metric algebras (classes defined by a set of metric equations) are exactly classes closed under (metric versions of) subalgebras, products and quotients. This result implies the class of normed vector spaces cannot be defined by metric equations, differently from the classical case where the class of vector spaces is equational. This phenomenon suggests that metric equations are not a very natural class of formulas since they cannot express such a typical class of metric algebras. Therefore we need a broader class of formulas to acquire an appropriate metric counterpart of the classical variety theory.

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Quasivarieties of Metric Algebras

Cited by 2 publications

References 2 publications

Continuous Varieties of Metric and Quantitative Algebras

Continuous Varieties of Metric and Quantitative Algebras

Varieties of Metric and Quantitative Algebras

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