On Complete Representations and Minimal Completions in Algebraic Logic, Both Positive and Negative Results

Fix a finite ordinal $n\geq 3$ and let $\alpha$ be an arbitrary ordinal. Let $\mathsf{CA}_n$ denote the class of cylindric algebras of dimension $n$ and $\sf RA$ denote the class of relation algebras. Let $\mathbf{PA}_{\alpha}(\mathsf{PEA}_{\alpha})$ stand for the class of polyadic (equality) algebras of dimension $\alpha$. We reprove that the class $\mathsf{CRCA}_n$ of completely representable $\mathsf{CA}_n$s, and the class $\sf CRRA$ of completely representable $\mathsf{RA}$s are not elementary, a result of Hirsch and Hodkinson. We extend this result to any variety $\sf V$ between polyadic algebras of dimension $n$ and diagonal free $\mathsf{CA}_n$s. We show that that the class of completely and strongly representable algebras in $\sf V$ is not elementary either, reproving a result of Bulian and Hodkinson. For relation algebras, we can and will, go further. We show the class $\sf CRRA$ is not closed under $\equiv_{\infty,\omega}$. In contrast, we show that given $\alpha\geq \omega$, and an atomic $\mathfrak{A}\in \mathsf{PEA}_{\alpha}$, then for any \(n/p