On bicompact semigroups
Katsumi Numakura · Institutional Repositories DataBase (IRDB) · 1952
We shall investigate in this note the structure of a minimal ideal of a bicompact semigroup, and to extend the theory of Suschkewitsch's kernel [1] (which he calls" Kerngruppe") of finite semigroups• to bicompact semigroupS.If S is a bicompact semigroup, then S has a minimal ideal K~ and K is completely simple in the sense of Rees [2] and is decomposed into groups which are isomorphic one another and have no element in common.Further, we shall show that minimal ideals (left~right and two-sided) of S are bicompact and closed in S. If S contains zero element, then K is zero alone, while if S has no zero, then K is a completely simple serpigroup without zero.1.A set S is called a semigroup, if in S a single-valued product ab is defined for every pair a, b of S such that for product the associative law holds:(ab)c = a(bc).By a sub-semigroup of S we mean a non-vacuous subset A of S with the property A 2 C A, i. e. ab E A for every a, b in A. By a left ideal of S we mean a non-vacuous subset L of S such that SLc L. Analogously~we can define a right ideal R of S. If M is both a left and a right ideal of S, then M is called a (two-sided) ideal of S.An element e of S is called an idempotent, if e 2 = e.An element o is termed zero, if Ox = 0 = xO for all x in S. Then it will easily be seen that the zero of S, if it exists, is uniquely defined and is anidemPOtent.An element 1 is termed the identity of S, if Ix = x = xl for all x in S. Then the identity of S, if it exists, is uniquely defined, and is an idempote.nt.2. If S is a semigroup and at the same time it is a topological space (in this note a topological space means a Hausdorff space), and mor~ver, the.multiplicative operation in the semigroup S is continuous in the topological space S, then S is called a topological semi• group.If a sub-semigroup T of S is closed (open) in the space S,. then we shall call T a closed (open) sub-semigroup etc.