Let S denote a Boolean ring with elements e, that is, a distributive, relatively complemented lattice with zero element 0 [2, p. 153]. In this paper we study real-valued functions which have a representation of the form

1.1

Any system S in which an addition is defined for some, but not necessarily all, pairs of elements can be imbedded in a natural way in a commutative semi-group G, although different elements in S need not always determine different elements in G (see §2). Theorem 2.1 gives necessary and sufficient conditions in order that a functional p(x) on S can be represented as the su prémuni of some family of additive functionals on S, and one such set of conditions is in terms of possible extensions of p(x) to G. This generalizes the case with 5 a Boolean ring treated by Lorentz [4], Lorentz imbeds the Boolean ring in a vector space and this could be done for the general S; but we prefer to imbed S in a commutative semi-group and to give a proof (see § 1) generalizing the classical Hahn-Banach theorem to the case of an arbitrary commutative semigroup.

A map wil be called k-chromatic or said to have chromatic number k if k is the least positive integer having the property that the countries of the map can be divided into k mutually disjoint (colour) classes in such a way that no two countries which have a common frontier line are in the same (colour) class.Heawood [4] proved that for h > 1 the chromatic number of a map on a surface of connectivity h is at most nh, where

In 1893 it was shown by Moore that the only commutative, associative division algebras with a finite number of elements are the well-known Galois fields [10, p. 220]. Twelve years later it was shown by Wedderburn that every associative division algebra with a finite number of elements is commutative [11], and hence a Galois field. It is conceivable that these results, particularly the theorem of Moore, motivated some of the work done by Dickson and published in two papers in 1906 [4;5]. The work referred to is an attempt to determine all commutative, non-associative1 division algebras with a finite number of elements.

Several papers have been written on the “new” multiplication of S-functions since Littlewood [3, p. 206] first suggested the problem. M. Zia-ud-Din [13] calculated the case {m} ⊗ {n} for mn ≤ 12, making use of the tables of the characters of the symmetric group of degree mn. Later Thrall [10,pp. 378-382] developed explicit formulae for the cases {m} ⊗ {2},{m} ⊗ {3}, {2} ⊗ {m} (where m is any integer).