Semi-lattice of varieties of quasigroups with linearity

Abstract

A \(\sigma\)-parastrophe of a class of quasigroups \(\mathfrak{A}\) is a class \({^{\sigma}\mathfrak{A}}\) of all \(\sigma\)-parastrophes of quasigroups from \(\mathfrak{A}\). A set of all pairwise parastrophic classes is called a parastrophic orbit or a truss. A parastrophically closed semi-lattice of classes is a bunch.

A linearity bunch is a set of varieties which contains the variety of all left linear quasigroups, the variety of all left alinear quasigroups, all their parastrophes and all their intersections. It contains 14 varieties, which are distributed into six parastrophic orbits. All quasigroups from these varieties are called dilinear. To obtain all varieties from the bunch, concepts of middle linearity and middle alinearity are introduced. A well-known identity or a system of identities which describes a variety from every parastrophic orbit of the bunch is cited. An algorithm for obtaining identities which describe all varieties from the parastrophic orbits is given. Examples of quasigroups distinguishing one variety from the other are presented.