For finite n\geq 3, and k\geq 4, the variety SNr_n\CA_{n+k} is not atom canonical

TL;DR

This paper demonstrates that for finite dimensions n≥3 and k≥4, certain algebraic structures are not atomically canonical, solving an open problem in algebraic logic by constructing specific counterexamples.

Contribution

It constructs an atomic representable polyadic equality algebra of finite dimension n≥3 whose completion's cylindric reduct is not in SNr_n ext{CA}_{n+4}, addressing a longstanding open problem.

Findings

Existence of atomic representable polyadic equality algebra for n≥3

Counterexample showing non-atom canonicity for k≥4

Solves an open problem in algebraic logic

Abstract

We show that there exists an atomic representable polyadic equality algebra of finite dimension n\geq 3, such that the cylindric reduct of its completion is not in SNr_n\CA_{n+4}, hence the result in the title. This solves an open problem in algebraic logic, though the values for k=n+1, n+2, n+3, is still, to the best of our knowlege unknown.