Comparison of outerplanarity and generalized outerplanarity properties for Cayley graphs of planar semigroups

We have found two infinite series of semigroups whose Cayley graphs have an outerplanarity property equivalent to the generalized outerplanarity property of their Cayley graphs, but not equivalent to the planarity property, and one infinite series of semigroups whose Cayley graphs have a generalized outerplanarity property equivalent to the planarity property of their Cayley graphs, but not equivalent to outerplanarity. It is proved that the Cayley graph of a finite semigroup is not isomorphic to any of the forbidden Sedláček's subgraphs by the characteristic property of generalized outer planarity with any orientation and edge coloring.