The set G[x] of polynomials over a group (G, + )—as well as the polynomial functions P(G) on (G, +) form near-rings with respect to addition and composition (substitution). See [1] for polynomials and [2] for near-rings. A number of results on G[x] can be deduced from [2].

Due to [1], the polynomials in G[x] can uniquely be represented in the following “normal form”:

The purpose of this paper is twofold: first to correct the statement of Theorem 1 in [4], and secondly to consider related problems in the class of ideally finite Lie algebras.

Throughout, L will denote a Lie algebra over a field K, F(L) will be its Frattini subalgebra and φ(L) its Frattini ideal. We will denote by the class of Lie algebras all of whose maximal subalgebras have codimension 1 in L. The Lie algebra with basis {u–1, u0, u1} and multiplication u–1u0 = u–1, u–1u1 = u0, u0u1 = u1 will be labelled L1(0).

By the rank r(S) of a finite semigroup S we shall mean the minimum cardinality of a set of generators ofS. For a group G, as remarked in [3], one has r(G)≦log2|G|, the bound being attained when G is an elementary abelian 2-group. By contrast, we shall see that there exist finite semigroups S for which r(S)≧|S| – 1. In the hope that it will not be considered too whimsical, we shall refer to a finite semigroup S of maximal rank (i.e. for which r(S) = |S|) as royal; a semigroup of next-to-maximal rank (i.e. for which r(S) = |S|–1) will be called noble.

Let G be a group having presentation

a braid relator and a commutator respectively, and define the graph of such a group as having vertices labelled x1,…, xn + 1 and such that xi and xj are joined if and only if rij = (xi, xj).

There are various algebras which may be associated with a discrete group G. In particular we may consider the complex group ring ℂG, the convolution Banach algebra l1(G), the enveloping C*-algebra C*(G) of l1(G), and the reduced C*-algebra determined by the completion of l1(G) under the left regular representation on l2(G). There is a substantial literature on the circle of ideas associated with the embeddings

In the following discussion we shall assume that pn≧0, qn≧0 for all n and that qn + 1 > qn → ∞. The (J, pn, qn) method of summation is defined as follows.

The series with the partial sum sn, is called summable (J, pn, qn) to s, and we write if the series

and converge to the sum functions p*(x) and p(s)(x) respectively for 0<x<1 and if τ(x) = p(s)(x)/p*(x)→s as x→1–0.

By a ring we shall mean an associative ring not necessarily containing an identity element. The fundamental definitions and properties of radicals may be found in Divinsky [2]. Similarly we refer to Howie [3] for the semigroup concepts.

If R is a ring Mn(R) will denote the ring of n × n matrices with entries from R. For many important radicals α it has been shown that α(Mn(R)) = Mn(α(R)) for all rings R and all positive integers n. However this is not the case for all radicals α. Associated with each radical α we define a set of positive integers S(α) by

Let l be an odd prime and let A be a commutative ring containing 1/l. Let K*(A;Z/lv) denote the mod lv algebraic K-theory of A [3]. As explained in [4] there exists a “Bott element” βv∈K21v–1(l–1)(Z[1/l];Z/lv) and, using the K-theory product we may, following [16, Part IV], form

which is defined as the direct limit of iterated multiplication by βv. There is a canonical localisation map

If a is an element of a complex unital Banach algebra whose numerical range is confined to a closed angular region with vertex at zero and angle strictly less than π, we imbed a in a holomorphic semigroup with parameter in the open right half plane.

There has been recently a great development in the theory of semigroups in Banach algebras (see [6]), with attention focused on the relation between the structure of a given Banach algebra and the existence of continuous or holomorphic non-trivial semigroups with certain properties with range in this algebra. The interest of this paper arises from the fact that we relate in it, we think for the first time, this new point of view in the theory of Banach algebras with the already classic one of numerical ranges [2,3]. The proofs of our results use, in addition to some basic ideas from numerical ranges in Banach algebras, the concept of extremal algebra Ea(K) of a compact convex set K in ℂ due to Bollobas [1] and concretely the realization of Ea(K) achieved by Crabb, Duncan and McGregor [4].

In previous papers [3, 4] the author has discussed the symmetric generalised Erdélyi–Kober operators of fractional integration defined by

where α>0, γ≧0 and the operators ℑiγ(η,α) and defined as in equations (1) and (2) respectively but with Jα−1, the Bessel function of the first kind replaced by Iα−1, the modified Bessel function of the first kind.

In this paper the “hypercore” of a semigroup S is defined to be the subsemigroup generated by the union of all the subsemigroups of S without non-universal cancellative congruences, provided that at least one such subsemigroup exists: otherwise it is taken to be the empty set. It is shown first that if the hypercore of S is nonempty (which holds, for example, when S contains an idempotent) then it is the largest subsemigroup of S with no non-universal cancellative congruence, is full and unitary in S, and is contained in the identity class of every group congruence on S (Theorem 1).

A semigroup is said to be congruence-free if and only if its only congruences are the universal relation and the identical relation. Congruence-free inverse semigroups were studied by Baird [2], Trotter [19], Munn [15,16] and Reilly [18]. In addition, results on congruence-free regular semigroups have been obtained by Trotter [20], Hall [4] and Howie [7].