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In recent developments in the algebraic theory of semigroups attention has been focussing increasingly on the study of congruences, in particular on lattice-theoretic properties of the lattice of congruences. In most cases it has been found advantageous to impose some restriction on the type of semigroup considered, such as regularity, commutativity, or the property of being an inverse semigroup, and one of the principal tools has been the consideration of special congruences. For example, the minimum group congruence on an inverse semigroup has been studied by Vagner [21] and Munn [13], the maximum idempotent-separating congruence on a regular or inverse semigroup by the authors separately [9,10] [15]. In this paper we study regular semigroups and our primary concern is with the minimum group congruence, the minimum band congruence and the minimum semilattice congruence, which we shall consistently denote by a, P and t] respectively.In § 1 we establish connections between /? and t\ on the one hand and the equivalence relations of Green [7] (see also Clifford and Preston [4, § 2.1]) on the other. If for any relation H on a semigroup S we denote by K* the congruence on S generated by H, then, in the usual notation, In § 2 we show that the intersection of a with jS is the smallest congruence p on S for which Sip is a UBG-semigroup, that is, a band of groups [4, p. 26] in which the idempotents form a unitary subsemigroup. The structure of such semigroups (and indeed of semigroups more general than this) has been investigated by Fantham [6]; his theorem (or rather the special case that is of interest here) is described below. A corollary of our result is that a n rj is the smallest congruence p for which Sjp is a USG-semigroup, that is, a semilattice of groups with a unitary subsemigroup of idempotents.These results lead naturally to a study of RU-semigroups (regular semigroups whose idempotents form a unitary subsemigroup), lSBG-semigroups (bands of groups whose idempotents form a subsemigroup) and SG-semigroups (semilattices of groups), and to the consideration of the minimum RU-congruence K, the minimum ISBG-congruence ( and the minimum SG-congruence ^ on a regular semigroup. The principal results of § 3 are that en/? = £ v K and a n tj = /; v K.In § 4 we show that any UBG-congruence on a regular semigroup can be expressed in a unique way as T ny, where x is a group congruence and y is a band congruence. A similar result holds for USG-congruences.

Munn [9] has shown that for a semigroup S satisfying the minimal condition on principal ideals, there is a natural one-to-one correspondence between irreducible representations of S and irreducible representations vanishing at zero of its 0-simple (or simple) principal factors; for the case of S finite, see Ponizovskii [11]. On the other hand, Clifford, [3] and [4], has obtained all representations of a completely 0-simple semigroup as "extensions" of those of its maximal subgroups. Combining their results, one can, in principle, obtain all irreducible representations of a semigroup satisfying the minimal conditions on principal left and right ideals and thus of finite semigroups. However, in constructing the representations of a completely 0-simple semigroup S=J(°(G;I, A;F), one has to solve the problem in matrix theory of factoring the block matrixwhere y is an irreducible representation of G (see [5, §5.4 ]).The main object of this paper is to show that, when dealing with finite semigroups and irreducible representations, it is possible to avoid the factorization problem and give explicit expressions for these representations. Let S be a finite semigroup and J a regular ^-class of S. By M¡ denote the Schützenberger representation of S by row-monomial matrices over G°, where G is the Schützenberger group of J (isomorphic to the maximal subgroups of S contained in /) ([5, § §2.4, 3.5], or [12]). For every x e S, let T(x) = y[Mj(x)], where y is a proper irreducible representation of G° by matrices over a field O, and y[Mj(x)] denotes the matrix over i> obtained by replacing each entry gKlt of M¡(x) by y(gAß). Then T is a representation of S by matrices over , and we prove (Theorem 1.7) that T has a unique nonnull irreducible constituent T* for which [T*(S)] = [T*(J)], where [r*(F)] denotes the linear closure of T*(F) (r* is given by (10)). The importance of this constituent Y* lies in the fact that every nonnull irreducible representation of S is equivalent to the constituent T* of some representation T relative to a suitable ./-class of S. This is an analogue to the well-known result in the theory of group representations : every irreducible representation of a group occurs as a constituent of the regular representation [1, 15.2]; this points to the fact that the direct sum of all Schützenberger representations of a semigroup is a suitable analogue of the right regular representation of a group. The proof depends essentially on an analogous property of finite

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Munn [9] has shown that for a semigroup S satisfying the minimal condition on principal ideals, there is a natural one-to-one correspondence between irreducible representations of S and irreducible representations vanishing at zero of its 0-simple (or simple) principal factors; for the case of S finite, see Ponizovskii [11]. On the other hand, Clifford, [3] and [4], has obtained all representations of a completely 0-simple semigroup as "extensions" of those of its maximal subgroups. Combining their results, one can, in principle, obtain all irreducible representations of a semigroup satisfying the minimal conditions on principal left and right ideals and thus of finite semigroups. However, in constructing the representations of a completely 0-simple semigroup S=J(°(G;I, A;F), one has to solve the problem in matrix theory of factoring the block matrix