Department of Mathematics
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For an algebra \( \mathbf{A} \), let \( p_n(\mathbf{A}) \) denote the number of \( n \)-ary operations which depend on all \( n \) variables. Then the \( p_n \)-sequence of \( \mathbf{A} \) is the integer sequence \[ p(A) = \lbrace p_0(\mathbf{A}), p_1(\mathbf{A}), \ldots, p_n(\mathbf{A}),\dots\rbrace. \] The basic open problem is, of course, to characterize all integer sequences which are representable as \( p_n \)-sequences of algebras. In this talk, we present one non-trivial example of a representable sequence which not only characterizes an affine variety based on abelian groups but also has the minimal extension property.

This research was done in collaboration with George Grätzer.