Semigroups that are the union of a group on $E^{3}$ and a plane

Abstract:

In Semigroups on a half-space, Trans. Amer. Math. Soc. 147 (1970), 1-53, Horne considers semigroups that are the union of a group $G$ and a plane $L$ such that $G \cup L$ is a three-dimensional half-space and $G$ is the interior. After proving a great many things about half-space semigroups, Horne introduces the notion of a radical and determines all possible multiplications in $L$ for a half-space semigroup with empty radical. (It turns out that $S$ has empty radical if and only if each $G$-orbit in $L$ contains an idempotent.) An example is provided for each configuration in $L$. However, no attempt was made to show that the list of examples actually exhausted the possibilities for a half-space semigroup without radical. Another way of putting this problem is to determine when two different semigroups can have the same maximal group. In this paper we generalize Horne’s results, for a semigroup without zero, by showing that if $S$ is any locally compact semigroup in which $L$ is the boundary of $G$, then $S$ is a half-space. Moreover, we are able to answer completely, for semigroups without radical and without a zero, the question posed above. It turns out that, with one addition (which we provide), Horne’s list of half-space semigroups without radical and without zero is complete.