Pure Maths Colloquium: Ashley Clayton

A countable subdirect power of a finite algebra \(A\) is a subalgebra of the countable cartesian power \(A^N\) consisting of countably many elements, for which the natural projection maps onto \(A\) are surjections. In 1982, McKenzie proved that for any finite non-abelian group \(G\), the number of non-isomorphic countable subdirect powers was uncountable, and is otherwise countable for finite abelian groups. In this talk, we take a tour through the case for some other algebras such as finite commutative semigroups with the aim of giving a McKenzie-like result, determining precisely those \(S\) which have countably many non-isomorphic subdirect powers.