A GENERALIZATION OF ADJAN'S THEOREM ON EMBEDDINGS OF SEMIGROUPS

In this paper we investigate the question of possibility to injectively map a semigroup into a group. Adjan's theorem provides a sufficient condition for such a map to exist for semigroups with relations l(i) = r(i), where both l(i) and r(i) are not empty. Presence of defining relations of the form l = 1 makes many combinatorial properties of semigroups significantly more complex. However, we generalize Adjan's theorem to the class of semigroups with defining relations of both kinds. We use Remmers's approach to exploit Van Kampen diagrams as major tool to abstract from the algebraic combinatorics behind the relations and, instead, work with more tangible objects, such as graphs on the plane.