Nilpotent algebras with maximal class in congruence modular varieties

Abstract

Ge eral representation theory by polynomials for nilpotent algebras in modular congruence varieties has been established and studied by several authors. An interesting open question is how to recursively construct nilpotent algebras with maximal class. In this thesis, several algebras are investigated: quaternary Mendelsohn quasigroups, Steiner quasigroups, Steiner loops, Steiner skeins, and p-groups. First, we obtain the structure theorem of finite nilpotent quaternary Mendelsohn quasigroups. By recursive construction we prove that for any natural number n, (i) there exist a nilpotent quaternary Mendelsohn quasigroup of order 4$\sp{\rm n + 1}$ with maximal class n, and a solvable quaternary Mendelsohn quasigroup of order 4$\sp{\rm n}$ with maximal class n; (ii) there exist a nilpotent Steiner quasigroup of order 3$\sp{\rm n + 1}$ with maximal class n, and a nilpotent Steiner loop of order 2$\sp{\rm n + 2}$ with maximal class n; (iii) any subdirectly irreducible quaternary Mendelsohn quasigroup (Steiner quasigroup) of class n + 1 with some conditions can be expanded to a subdirectly irreducible quaternary Mendelsohn quasigroup (Steiner quasigroup) of class n + 2. We also give a new and simple recursive construction for a nilpotent Steiner skeins of order 2$\sp{\rm n + 2}$ with maximal class n such that all its derived Steiner loops are of nilpotence class n; and we represent by polynomials the dihedral group and the generalized quaternion group of order 2$\sp{\rm n +1}$ which are nilpotent p-groups with maximal class n.