Approval compatible voting rules

Suppose voters are asked to submit approval ballots for a certain set of alternatives, with approval voting applied to determine a winning alternative. The same voters are then asked to report rankings over these alternatives, and some voting rule intended for ranked ballots is applied. If voters are sincere, can an approval winner possibly win this second election? Can an approval loser lose that election, or all approval co-winners be co-winners of the election? These questions give rise to three notions of approval compatibility for voting rules: positive, negative, and uniform positive approval compatibility (PAC, NAC, and UPAC). We find that NAC is a very weak notion and UPAC is a very strong one. Moreover, PAC, a stronger variant of it called OPAC, and a weaker variant of UPAC called FUPAC divide usual voting rules into four families: Condorcet-consistent rules satisfy all of them; K-approval rules for K >= 2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$K\ge 2$$\end{document} satisfy none; plurality, plurality with runoff and STV satisfy OPAC but fail FUPAC; and Borda satisfies FUPAC and PAC but fails OPAC.