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Sparks, Athena

On the number of clonoids. (English) Zbl 1436.08005

Algebra Univers. 80, No. 4, Paper No. 53, 10 p. (2019).
A clonoid \(C_{A,B}\) is a set of finitary functions from a set \(A\) to a set \(B\) that is closed under taking minors. The author studies the size of clonoids for finite sets and algebras. For any finite set \(A\) and any two-element algebra \(B\), \(C_{A,B}\) is finite iff \(B\) has an NU-term, it is countably infinite iff \(B\) has a Mal’cev term but no majority term and it has size continuum otherwise. If \(A\) is a finite set and \(B\) a finite idempotent algebra then \(C_{A,B}\) has size continuum iff \(B\) has no cube term. If \(B\) has a cube term, then there are countably many such clonoids.
Reviewer: Ivan Chajda (Přerov)

Cited in 1 Review
Cited in 3 Documents

MSC:

08A40 Operations and polynomials in algebraic structures, primal algebras
06E30 Boolean functions

Keywords:

clones; polymorphisms; Boolean functions; minors; clonoids; NU-function; cube function
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\textit{A. Sparks}, Algebra Univers. 80, No. 4, Paper No. 53, 10 p. (2019; Zbl 1436.08005)
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