## On the number of clonoids.(English)Zbl 1436.08005

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.

### MSC:

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

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