Neat embeddings as adjoint situations

Looking at the operation of forming neat α\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document}-reducts as a functor, with α\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document} an infinite ordinal, we investigate when such a functor obtained by truncating ω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega $$\end{document} dimensions, has a right adjoint. We show that the neat reduct functor for representable cylindric algebras does not have a right adjoint, while that of polyadic algebras is an equivalence. We relate this categorial result to several amalgamation properties for classes of representable algebras. We show that the variety of cylindric algebras of infinite dimension, endowed with the merry go round identities, fails to have the amalgamation property answering a question of Németi’s. We also study two variants of the so-called cylindric polyadic algebras introduced by Ferenczi (all are reducts of polyadic equality algebras, that are also varieties). We show that one is more cylindric than polyadic, and that the other is more polyadic than cylindric. Our classification is determined by results on neat embeddings and amalgamation expressed from the point of view of category theory, thereby witnessing, and, indeed, further emphasizing, the dichotomy between the cylindric and polyadic paradigms. For example, the first class does not have the unique neat embedding property, fails to have the amalgamation property and the neat reduct functor does not have a right adjoint, while the second class has the unique neat emdedding property, the superamalgamation property and the neat reduct functor is strongly invertible. Other results, like first order definability of the class of neat reducts and the class of completely representable algebras, confirming our classification along these lines are presented.