Polynomial Completeness in Algebraic Systems

Kaarli, Kalle; Pixley, Alden F.

doi:10.1201/9781482285758

Cited by 61 publications

(80 citation statements)

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“…[43] for background, recalling only that an algebra A is affine complete if every congruence-compatible operation on A is a polynomial function of A and that a variety is affine complete if every member is affine complete. Classic examples are the variety of Boolean algebras, and more generally any arithmetical variety generated by a finite set of primal algebras [43, p. 158].…”

Section: Varieties Of Lattices and Lattice-based Algebrasmentioning

confidence: 99%

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Natural extensions and profinite completions of algebras

et al. 2011

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This paper investigates profinite completions of residually finite algebras, drawing on ideas from the theory of natural dualities. Given a class A = ISP(M), where M is a set, not necessarily finite, of finite algebras, it is shown that each A ∈ A embeds as a topologically dense subalgebra of a topological algebra n A (A) (its natural extension), and that n A (A) is isomorphic, topologically and algebraically, to the profinite completion of A. In addition it is shown how the natural extension may be concretely described as a certain family of relation-preserving maps; in the special case that M is finite and A possesses a single-sorted or multisorted natural duality, the relations to be preserved can be taken to be those belonging to a dualising set. For an algebra belonging to a finitely generated variety of lattice-based algebras, it is known that the profinite completion coincides with the canonical extension. In this situation the natural extension provides a new concrete realisation of the canonical extension, generalising the well-known representation of the canonical extension of a bounded distributive lattice as the lattice of up-sets of the underlying ordered set of its Priestley dual. The paper concludes with a survey of classes of algebras to which the main theorems do, and do not, apply.

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Section: Varieties Of Lattices and Lattice-based Algebrasmentioning

confidence: 99%

“…Classic examples are the variety of Boolean algebras, and more generally any arithmetical variety generated by a finite set of primal algebras [43, p. 158]. (For a full discussion of the close relationship between arithmeticity and affine completeness, see [43]. )…”

Section: Varieties Of Lattices and Lattice-based Algebrasmentioning

confidence: 99%

Natural extensions and profinite completions of algebras

et al. 2011

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“…An algebra is called affine complete if every congruence preserving function is a polynomial function (cf. [10]). In [8], Idziak and S lomczyńska considered those functions which, in addition to being congruence preserving, also preserve certain local structures that can be associated with the prime quotients in the congruence lattice of the algebra.…”

Section: Introductionmentioning

confidence: 99%

Types of polynomial completeness of expanded groups

Aichinger

Mudrinski

2009

Algebra univers.

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The polynomial functions of an algebra preserve all congruence relations. In addition, if the algebra is finite, they preserve the labelling of the congruence lattice in the sense of Tame Congruence Theory. The question is for which algebras every congruence preserving function, or at least every function that preserves the labelling of the congruence lattice, is a polynomial function. In this paper, we investigate this question for finite algebras that have a group reduct.

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“…Affine completeness has been investigated for various kinds of algebras. (See [7] for a survey.) In general, however, there are non-polynomial compatible functions and our aim is to investigate this phenomenon and characterize the compatible functions.…”

Section: Introductionmentioning

confidence: 99%

“…The variety K of Kleene algebras is characterized in K ∨ S by the identity x = x * * (7) and the variety S of Stone algebras is characterized in K ∨ S by the identity…”

Section: Introductionmentioning

confidence: 99%