Direct products of cyclic semigroups allowing outerplanar cayley graphs and their generalizations

The characteristic properties of outerplanarity and generalized outerplanarity of Cayley graphs of direct products of cyclic semigroups are proved in terms of copresentations. The main idea of the proof of the theorems given in the article is the following: if the conditions discovered as a result of the study are met, then the semigroup admits a generalized outer-plane [respectively, outer-plane] layout of its Cayley graph (that is, such a layout in which each edge belongs to one face of at least one of its vertices, and the edges do not intersect in the plane) [accordingly, such a layout in which all the vertices belong to the same face, and the edges do not intersect in the plane]; conversely, according to the law of contraposition, if the found conditions are not met, then a subgraph is indicated that is homeomorphic to one of the forbidden configurations. The reasoning is carried out by analogy with the study of semigroups admitting planar graphs, while the forbidden configurations are changed to new ones, due to the Chartrand-Harari and Sedlacek criterion.