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On the number of finite algebraic structures

Erhard Aichinger, Peter Mayr, R. McKenzie (2014)

Journal of the European Mathematical Society

We prove that every clone of operations on a finite set A , if it contains a Malcev operation, is finitely related – i.e., identical with the clone of all operations respecting R for some finitary relation R over A . It follows that for a fixed finite set A , the set of all such Malcev clones is countable. This completes the solution of a problem that was first formulated in 1980, or earlier: how many Malcev clones can finite sets support? More generally, we prove that every finite algebra with few...

Orderings of monomial ideals

Matthias Aschenbrenner, Wai Yan Pong (2004)

Fundamenta Mathematicae

We study the set of monomial ideals in a polynomial ring as an ordered set, with the ordering given by reverse inclusion. We give a short proof of the fact that every antichain of monomial ideals is finite. Then we investigate ordinal invariants for the complexity of this ordered set. In particular, we give an interpretation of the height function in terms of the Hilbert-Samuel polynomial, and we compute bounds on the maximal order type.

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