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 dimension \(n\) \(\sf RA\) relation algebras. \(\mathbf{PA}_{\alpha}(\mathsf{PEA}_{\alpha})\) stand for polyadic (equality) \(\alpha\). We reprove that \(\mathsf{CRCA}_n\) completely representable \(\mathsf{CA}_n\)s, CRRA\) \(\mathsf{RA}\)s are not elementary, result Hirsch Hodkinson. extend this to any variety V\) between diagonal free \(\mathsf{CA}_n\)s. show strongly in is elementary either, reproving Bulian For algebras, we can will, go further. closed under \(\equiv_{\infty,\omega}\). In contrast, given \(\alpha\geq \omega\), atomic \(\mathfrak{A}\in \mathsf{PEA}_{\alpha}\), then \(n<\omega\), \(\mathfrak{Nr}_n\mathfrak{A}\) \(\mathsf{PEA}_n\). certain reducts \(\mathsf{PA}_{\alpha}\)s, happen varieties, elementary. polyadic-cylindric \(\alpha\), introduced by Ferenczi, (slightly altering representing algebras) coincide with ones. last cylindrifications commute only one way, sense weaker than full fledged commutativity enjoyed classical Finally, address closure Dedekind-MacNeille completions cylindric-like \(\mathsf{PA}_{\alpha}\)s infinite ordinal, proving negative results first positive ones second.