Primitively Recursive Categoricity for Unars and Equivalence Structures

This continues the study of the primitively recursive categoricity of structures for which there exists a primitively recursive decision algorithm with witnesses of all \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$ \Sigma $\end{document}-formulas. Considering the equivalence structures, we find a complete criterion for primitively recursive categoricity over the class \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$ K_{\Sigma} $\end{document}, which coincides with the already known criterion for computable categoricity. As regards unars, the structures with one arbitrary unary function, we distinguish some conditions for primitively recursive categoricity over \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$ K_{\Sigma} $\end{document} and also for the absence of this categoricity. In particular, we find a full description of primitively recursive injective unars categorical over \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$ K_{\Sigma} $\end{document}.