The hulls of $C(Y)$
Marlow Anderson, Paul F. Conrad · Rocky Mountain Journal of Mathematics · 1982
Introduction.Let C(Y) be the set of all continuous real-valued functions on a completely regular space Y. Then C(Y) can be considered as an /-group Gi or as a semiprime ring (73, and in each case it admits various X-hulls, which are minimal essential extensions with some property X.We show that <7f is essentially the same as Gf and investigate the structure of these Z-hulls.All of these hulls are contained in the complete ring of quotients Q(Y) of G 3 , and, in fact, Q(Y) is the lateral completion of Gì or of G 3 .In the first two sections we summarize the theory known for abelian /-group and commutative semiprime ring X-hulls.The third section contains a description of the hulls of C(F), and their relationships with one another.§4 contains characterizations of C(Y) considered as an abstract /-group.For further information about lattice-ordered groups (/-groups), see [9] or [14]; for semiprime rings, see [26]; for C(F), see [24].We will use £7^ (II7^) to represent the restricted (unrestricted) direct product of the groups or rings T x \ in the case of /-groups, these groups are equipped with the cardinal order.We wish to acknowledge the valuable advice of Jack Porter about the topological results that appear in this paper.In particular, Theorem 3.9 and Example 3.12 are entirely due to him.