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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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