Semigroups acting on continua

J. M. Day, A. D. Wallace
1967 Journal of the Australian Mathematical Society  
A semigroup is a nonvoid Hausdorff space together with a continuous associative multiplication. (The latter phrase will generally be abbreviated to CAM and the multiplication in a semigroup will be denoted by juxtaposition unless the contrary is made explicit.) Any Hausdorff space may be supplied with a CAM, and, for example, one may define xy = x for all x and y. The addition of algebraic conditions may change the situation greatly and a circle together with a diameter does not admit a CAM
more » ... not admit a CAM with unit. It was shown in [W 1] (see [KW 1] for another example) that the space consisting of the curve y = sin (I/a;), 0 < x ^ 1, together with its limit continuum, does not admit a CAM with unit. (This result follows readily from a result of Robert Hunter's [H].) An act is such a continuous function TxX-^X that T is a semigroup and X is a nonvoid Hausdorff space and, denoting the value of the anonymous function at the place (t, x) by tx, the associativity condition MM) = (hh)* holds for all t^t^eT and all x e X. We shall refer to this situation as an action of T on X and say that T acts on X, or use similar terminology. Again, any semigroup may act upon any space, for example one may put tx = x for all t e T and all x e X. Moreover, the situation in which T is a group is so well known as not to require explication. However, when T is merely a semigroup, very little is known without additional conditions on T and X of an algebraic and metric nature, and it is our intention here to inaugurate such an investigation, of a modest character. Put in its simplest form, we shall give conditions under which a compact connected semigroup may not act upon the sinuscurve described in an earlier paragraph. In more detail, suppose that the space X contains an open dense half-line whose complement is a set C, that there is some q e X such that Tq -X, that T acts unitarily on X (x e Tx for each xeX), and that a
doi:10.1017/s1446788700004171 fatcat:vvdkjkrvgjdmdgs2qymefn7ki4