Planarity ranks of semigroup varieties generated by semigroups of order four

This paper classifies semigroup varieties generated by fourth-order semigroups according to their planarity ranks. The aim of the study is to establish a complete list of possible values of planarity ranks  and to identify the main factors determining the possibility of planar stacking of Cayley graphs of free semigroups of the considered varieties. Methods from graph theory and algebras of identities are applied, using innovative algorithmic approaches to verify equality via the automated proof systems Prover9 and Mace4. The existing flat graph stackings for the Cayley graphs of the semigroups under consideration are shown in the figures. If there is no planarity, the particular forbidden minor discovered is indicated: a complete fifth-order graph or a complete bipartite graph containing three vertices in each of the parts. Special attention is paid to the statistical processing of the obtained results by the principal components analyse and the construction of hierarchical clustering. The figures show hierarchical trees, factor planes, correlation circles, and column diagrams of general inertia decomposition along coordinate axes. Although the planarity of the Cayley graph for a free semigroup of a manifold was previously intuitively associated with the complexity degree of the defining identities, in this paper this dependence is for the first time given a rigorous quantitative expression, depicted in tables. Within the framework of the study, auxiliary parameters are introduced, which allows to significantly increase the explanatory power of the model and divide manifolds into groups according to topological characteristics. As a result of the analysis it is established that the leading factors influencing the value of ranks are the parameters reflecting the differences of positions of the symbol «z» in the basis identities.