Conditions for embeddability of semigroups in groups (Q1905260)

Let \(K^q_p\) denote the class of all semigroups \(S= \langle x_1, \dots, x_n\mid A_i= B_i\), \(i\in I\rangle\), where the defining words satisfy the small cancellation conditions \(C_s (p)\) and \(D(q)\). \textit{E. V. Kashintsev} proved [in Int. J. Algebra Comput. 2, No. 4, 433-441 (1992; Zbl 0765.20028)]\ that semigroups of the classes \(K^2_4\) and \(K^3_3\) are embeddable in groups and constructed there examples of semigroups of the class \(K_2^q\) for \(q>2\) that are not embeddable in groups. Here, the author answers two questions stated in the paper mentioned above. Theorem 1. All semigroups of the class \(K^2_3\) are embeddable in groups. Theorem 2. For any \(q>2\), there exist semigroups with cancellation that belong to the class \(K^q_2\) and are not embeddable in a group. Then the author generalizes Theorem 1 to a more general case of corepresentations (Theorem 3) and gives an addition to Theorem 2 (Theorem 4) which concerns the semigroups that satisfy the left (right) cancellation condition and are embeddable in a group.