Finitely generated equational classes

Aichinger, Erhard; Mayr, Peter

doi:10.1016/j.jpaa.2016.01.001

Cited by 17 publications

(35 citation statements)

References 15 publications

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“…By Theorem 5.3 in [1], C A,B satisfies the DCC. Hence there exists m ∈ N such that C m = C n for all n ≥ m. By (3.1), C = C m and C is finitely related.…”

Section: Cube Termmentioning

confidence: 92%

On the number of clonoids

Sparks

2019

Algebra Univers.

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A clonoid is a set of finitary functions from a set A to a set B that is closed under taking minors. Hence clonoids are generalizations of clones. By a classical result of Post, there are only countably many clones on a 2-element set. In contrast to that, we present continuum many clonoids for A = B = {0, 1}. More generally, for any finite set A and any 2-element algebra B, we give the cardinality of the set of clonoids from A to B that are closed under the operations of B. Further, for any finite set A and finite idempotent algebra B without a cube term (with |A|, |B| ≥ 2) there are continuum many clonoids from A to B that are closed under the operations of B; if B has a cube term there are countably many such clonoids.

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“…By Theorem 5.3 in [1], C A,B satisfies the DCC. Hence there exists m ∈ N such that C m = C n for all n ≥ m. By (3.1), C = C m and C is finitely related.…”

Section: Cube Termmentioning

confidence: 92%

On the number of clonoids

Sparks

2019

Algebra Univers.

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“…Then every (F p , F q )linearly closed clonoid C is generated by its unary functions. Thus C = Clg(C [1] ).…”

Section: Definition 42mentioning

confidence: 99%

“…The aim of this paper is to describe sets of functions from F q to F p that are closed under all linear mappings from the left and from the right, in the case p and q are powers of distinct primes. We are dealing with sets of functions with different domains and codomains; such sets are investigated, e.g., in [1] and are called clonoids. Let B be an algebra, and let A be a non-empty set.…”

Section: Introductionmentioning

confidence: 99%

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Closed sets of finitary functions between finite fields of coprime order

Fioravanti

2020

Algebra Univers.

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We investigate the finitary functions from a finite field $$\mathbb {F}_q$$ F q to the finite field $$\mathbb {F}_p$$ F p , where p and q are powers of different primes. An $$(\mathbb {F}_p,\mathbb {F}_q)$$ ( F p , F q ) -linearly closed clonoid is a subset of these functions which is closed under composition from the right and from the left with linear mappings. We give a characterization of these subsets of functions through the invariant subspaces of the vector space $$\mathbb {F}_p^{\mathbb {F}_q\backslash \{0\}}$$ F p F q \ { 0 } with respect to a certain linear transformation with minimal polynomial $$x^{q-1} - 1$$ x q - 1 - 1 . Furthermore we prove that each of these subsets of functions is generated by one unary function.

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“…Let A be a set, and let B be an algebra. Following [AM16], a subset C of n∈N B A n is called a clonoid from A to B if C is closed under taking minors of functions, and for every k ∈ N, the set…”

Section: Definable Setsmentioning

confidence: 99%