On the lattice of varieties of bands of groups
THOMAS E. HALL, Peter R. Jones · Pacific Journal of Mathematics · 1980
1* Introduction* When considered as semigroups with an additional unary operation x —> where x~ denotes the (unique) inverse of x in the subgroup to which it belongs, the class CR of completely regular semigroups (often called unions of groups) forms a variety of universal algebras, containing as a subvariety thevariety BG of bands of groups (those completely regular semigroups on which J%f is a congruence) ([12]). In this paper results of Spitznagel [14] on the lattice of congruences on a band of groups are applied to show that T*(BG), the lattice of subvarieties of BG, is modular (Theorem 3.1). Petrich [12, 13] considered various subvarieties of BG but left open the problem [13, p. 1196] of finding the join of the subvarieties B and CS (of bands and of completely simple semigroups respectively). We show that B V CS = POBG, the variety of pseudo-orthodox bands of groups, and is thus strictly contained in BG. (If V is a variety of completely regular semigroups and SeCR we shall call S pseudo-V if eSe e V for every idempotent e of S.) This result is actually an immediate corollary to our characterization of the join O V NBG of the varieties of orthodox completely regular semigroups and of normal bands of groups. Theorem 3.1 is also applied to directly decompose various sublattices of Ψ*{BG).