### Harmonic inverse semigroups

Mario Petrich, Stuart A. Rankin
1989 Glasgow Mathematical Journal
Introduction and summary. An inverse semigroup 5 shall be said to be harmonic if every congruence on 5 is determined by any one of its classes. In other words, if A and p are congruences on 5 having a congruence class in common, then A = p. The class & of all harmonic semigroups contains all bisimple inverse semigroups, as proved by Zitomirskii [11] and also by Schein [10], and all congruence-free inverse semigroups. Moreover, & is contained in the class of all 0-simple or simple inverse
more » ... mple inverse semigroups, as is easy to see. We shall show that there exist non-bisimple, non-congruence-free harmonic semigroups and that there are simple inverse semigroups which are not harmonic. The results of this paper are grouped as follows. Section 2 is a preliminary one, containing notation and a brief discussion of relevant facts. In Section 3, we investigate the implications of two congruences having a congruence class in common. One interesting and useful observation is that two congruences having the same trace and a class in common must have an idempotent class in common. The elementary fact that any two group congruences with a class in common must be equal is essentially a consequence of this observation. Harmonic semigroups are discussed in Section 5. The notions of kernel-harmonic and trace-harmonic semigroups are introduced and studied in Sections 6 and 7 respectively. As mentioned earlier, simple inverse semigroups are central to these investigations. One manageable class of simple inverse semigroups consists of the Bruck extensions of inverse monoids. Section 8 contains a characterization of harmonic Bruck extensions, as well as several additional results which are illuminating in view of the various conditions studied in Sections 6 and 7. It is in Section 8 that we construct a class of harmonic semigroups which properly contains the class of all bisimple or congruence-free inverse semigroups. The characterization theorem also provides a ready source of examples of simple non-harmonic inverse semigroups, thus showing that fe is indeed properly contained in the class of all (O-)simple inverse semigroups. In particular, we are able to answer a question posed by Schein in [10]: it is not true that every congruence contained in S on a (O-)simple inverse semigroup is determined by each of its congruence classes. Theorem 3.4 and the discussion in Section 4 provide considerable information about the extent to which a congruence is determined by a given congruence class. We conclude the paper with a brief discussion in Section 9 of £-unitary inverse semigroups in the context of harmonicity. The discussion follows the general theme developed by Reilly in [9]. Notation and relevant facts. Throughout this paper, S will denote an inverse semigroup and E its semilattice of idempotents. For any subset A of S, we shall write E A for EH A.