Relative lengths of Maltsev conditions

Abstract

We solve some problems about relative lengths of Maltsev conditions, in particular, we provide an affirmative answer to a classical problem raised by A. Day more than 50 years ago. In detail, both congruence distributive and congruence modular varieties admit Maltsev characterizations by means of the existence of a finite but variable number of appropriate terms. A. Day showed that from Jónsson terms t0,…,tn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$t_0,\\ldots , t_n$$\\end{document} witnessing congruence distributivity it is possible to construct terms u0,…,u2n-1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$u_0,\\ldots , u _{2n-1} $$\\end{document} witnessing congruence modularity. We show that Day’s result about the number of such terms is sharp when n is even. We also deal with other kinds of terms, such as alvin, Gumm, directed, as well as with possible variations we will call “specular” and “defective”. All the results hold when restricted to locally finite varieties.