Delphic Semigroups by David G. Kendall

an array is said to converge to an element u when the ith marginal product converges to u\ an element is said to be infinitely divisible when it possesses a Jfeth root for each fee2. (A) There exists a continuous homomorphism A from the semigroup into the additive semigroup of nonnegative reals, such that A(u) = 0 if and only if u is the neutral element. (B) The set {u':u'\u} of factors of any given element u is compact. (C) If a triangular array converges to u, and if the array satisfies the condition A(u(i, i))—>0 as i—><*> uniformly for l ^ j r g i , then u is infinitely divisible. As a nontrivial example we mention here only the multiplicative semigroup of positive renewal sequences; for the complete details of this and other examples, as well as for the proofs of the following theorems, reference should be made to [ l ] and [2]. The first property of a delphic semigroup is that every infinitely divisible element u can be represented as a limit as in (C). Next, it can be shown that the elements of such a semigroup can be partitioned into three exhaustive and mutually exclusive classes; the elements in the first class are indecomposable; those in the second class are decomposable but possess an indecomposable factor; those in the third class are infinitely divisible and possess no indecomposable factor. Finally it can be shown that an arbitrary element of such a semigroup possesses at least one representation in the form