Semisimple Bands
B. D. Arendt · Transactions of the American Mathematical Society · 1969
A band is a semigroup in which every element is idempotent.A right congruence t on a semigroup 5 is said to be modular if there exists an element e in S such that (ex)TX for all x in S. The symmetric definition holds for left congruences, and a two-sided congruence is said to be modular if it is modular both as a left and a right congruence and hence has a two-sided identity.Following Oehmke [7] (see also [3]) we define the radical Rr(Ru Rt) of a band S to be the intersection of all the maximal, modular right (left, two-sided) congruences on S. Sis said to be x-semisimple (x-radical) if the radical Rx is the identity relation í (universal relation v) on S.There are three distinct types of maximal, modular right congruences on a band.This classification enables us to prove that a band is r-semisimple if and only if it is a semilattice r of right zero semigroups Sy such that for x, y e Sy, x^v, the principal left ideals Sx and Sy are disjoint.A band has only one type of maximal, modular two-sided congruence, and this is used to show that a band S is r-semisimple if and only if it is a semilattice.Kimura [5, Lemma 1] has characterized a rectangular band as the cartesian product of two sets with multiplication given by (a, b)(c, d) = (a, d).The structure theory for arbitrary bands begins with the following result by McLean [6, Theorem