Double ideals in compact semirings

Abstract

Communicated by G. B. PrestonBy a topological semiring we mean a Hausdorff space S together with two continuous associative operations on S such that one (called multiplication) distributes across the other (called addition). That is, we insist that x(y + z) = xy + xz and (x + y) z = xz + yz for all x,y and z in S. Note that, in contrast to the purely algebraic situation [1,2], we do not postulate the existence of an additive identity which is a multiplicative zero.In this note we point out conditions under which the existence of such an element is equivalent to the double simplicity of the semiring. We also discuss maximal and minimal double ideals together with several examples.We For references on the properties of these sets, the reader may see [4,5]. We say a set / is a double ideal if it is both an additive and a multiplicative ideal. A semiring containing no proper double ideal will be called doubly simple.

Product structure in topological algebras

“…We remark in passing that from Lemma 10 come maximal double T-ideals, which are open, as first noted in Robbie (1970) and later, case T = M, in Bertman and Selden (1973).…”

“…Note that this lemma appears first in Robbie(1970),and later, in the special case T = M in Bertman and Selden (1973). We will need it in a different special case namely, R = M x M and T = A, when M is itself a semiring.…”