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