Theorems on compact totally disconnected semigroups and lattices
Katsumi Numakura · Proceedings of the American Mathematical Society · 1957
katsumi numakuraIn the theory of topological algebras, it is well-known that a compact totally disconnected group (or a compact totally disconnected ring) is a projective limit of finite groups (or finite rings).In §2 of this paper we shall establish a theorem on compact totally disconnected Hausdorff semigroups which is similar to that for compact totally disconnected groups.The theorem can be applied to other compact totally disconnected algebraic systems.For example, it can be applied to compact totally disconnected rings, compact totally disconnected distributive lattices, etc.In §3, we shall prove an imbedding theorem for compact totally disconnected distributive lattices which is well-known in the discrete case [l].2Throughout the paper we will use the terminology of Wallace [3; 4] and [5].Thus a mob S is a Hausdorff space together with a continuous associative multiplication.A ES is a T-ideal, where T is a nonvacuous subset of S, if and only if TA EA and A TEA.Moreover, a topological lattice L is a Hausdorff space together with a pair of continuous functions A-LXL->L, V: LXL-+L satisfying the usual conditions for lattice operations.The author wishes to acknowledge the advice and helpful suggestions of Professor A. D. Wallace in the preparation of this paper.