On the Number of Clones Containing All Constants (A Problem of R. McKenzie)

Ágoston, I.; Demetrovics, János; Hannák, L.

doi:10.1016/b978-0-444-87759-8.50006-7

Cited by 12 publications

(11 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…We will investigate those clones that contain all constant functions; such clones have been called constantive in [14]. From [2] we know that, if |A| ≥ 3, there are 2 ℵ 0 clones containing all constant functions on A. In [14] it was proved that for |A| ≥ 4 infinitely many clones on A contain a ternary Mal'cev operation.…”

Section: Introductionmentioning

confidence: 99%

Polynomial clones on groups of order pq

Aichinger

Mayr

2006

Acta Math Hung

View full text Add to dashboard Cite

show abstract

Section: Introductionmentioning

confidence: 99%

Polynomial clones on groups of order pq

Aichinger

Mayr

2006

Acta Math Hung

View full text Add to dashboard Cite

show abstract

“…Although the foregoing exhibits a simple structure of the lattice polynomial clones over {0, 1}, there are uncountably many polynomial clones over n when n ≥ 3 [1]. So the structure of the polynomial clones over n when n ≥ 3, although still a lattice with minimum and maximum element, is very complex and still largely a mystery.…”

Section: Proof Supposementioning

confidence: 99%

Polynomial clone reducibility

Culver

2013

Arch. Math. Logic

View full text Add to dashboard Cite

Polynomial-clone reducibilities are generalizations of the truth-table reducibilities. A polynomial clone is a set of functions over a finite set X that is closed under composition and contains all the constant and projection functions. For a fixed polynomial clone C, a sequence B ∈ X ω is C-reducible to A ∈ X ω if there is an algorithm that computes B from A using only effectively selected functions from C. We show that if A is a Kurtz random sequence and C1 C2 are distinct polynomial clones, then there is a sequence B that is C1-reducible to A but not C2-reducible to A. This implies a generalization of a result first proved by Lachlan for the case |X| = 2. We also show that the same result holds if Kurtz random is replaced by Kolmogorov-Loveland stochastic.

show abstract

“…On a two-element set, there are exactly 7 polynomially inequivalent algebras. However, by [1], if A is finite and |A| ≥ 3, then there are 2 ℵ 0 polynomially inequivalent algebras on A. In this paper, we restrict our attention to algebras with a Mal'cev term (Mal'cev algebras for short).…”

Section: Introductionmentioning

confidence: 99%