The Structure of Idempotent Translatable Quasigroups

We prove the main result that a groupoid of order n is an idempotent and k-translatable quasigroup if and only if its multiplication is given by x·y=(ax+by)(modn)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x\cdot y=(ax+by)(\mathrm{mod}\,n)$$\end{document}, where a+b=1(modn)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a+b=1(\mathrm{mod}\,n)$$\end{document}, a+bk=0(modn)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a+bk=0(\mathrm{mod}\,n)$$\end{document} and (k,n)=1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(k,n)=1$$\end{document}. We describe the structure of various types of idempotent k-translatable quasigroups, some of which are connected with affine geometry and combinatorial algebra, and their parastrophes. We prove that such parastrophes are also idempotent k-translatable quasigroups and determine when they are of the same type as the original quasigroup. In addition, we find several different necessary and sufficient conditions making a k-translatable quasigroup quadratical.