The neat embedding problem for algebras other than cylindric algebras and for infinite dimensions (Q2921030)

This paper extends the solution to Monk's neat embedding problem from finite-dimensional cylindric algebras to (possibly infinite-dimensional) quasi-polyadic equality algebras, quasi-polyadic algebras, and substitution algebras. In the first part, for integers \(n>m>2\) and linear ordering \(r\) the authors construct an \(m\)-dimensional quasi-polyadic equality algebra \(C(m,n,r)\). \(C(m,n,r)\) is a neat \(m\)-dimensional reduct of the \(n\)-dimensional quasi-polyadic equality algebra \(C(n,n,r)\). Let \(K\) be the class of subalgebras of neat \(m\)-dimensional reducts of \(n+1\)-dimensional quasi-polyadic equality algebras. For finite \(r\), \(C(m,n,r)\) is not in \(K\). If \(r\) has an infinite ascending sequence then \(C(m,n,r)\) is elementarily equivalent to an algebra in \(K\). These properties are used to show that \(K\) is not definable by any finite set of first-order conditions within the class of subalgebras of neat \(m\)-dimensional reducts of \(n\)-dimensional quasi-polyadic equality algebras. Similar results for infinite \(n\) are obtained in the second part of the paper.