Metrizability of Topological Semigroups on Linearly Ordered Topological Spaces

The authors use techniques and results from the theory of generalized metric spaces to give a new, short proof that every connected, linearly ordered topological space that is a cancellative topological semigroup is metrizable, and hence embeddable in R. They also prove that every separable, linearly ordered topological space that is a cancellative topological semigroup is metrizable, so embeddable in R. 1. Background and Introduction A linearly ordered topological space (LOTS) L is a linearly ordered set L with the open interval topology. A cancellative topological semigroup on L is a semigroup with a continuous semigroup operation such that ab = ac, ba = ca and b = c are equivalent for any a, b, c ∈ L. A question that can be traced to Abel and Lie, and was listed as the second half of Hilbert’s fifth problem, essentially asks whether a cancellative topological semigroup on a connected, linearly ordered topological space can be embedded in the real line. The history of the problem, and the various solutions and partial solutions and related questions, are most thoroughly documented by Hofmann and Lawson [7]. A very abbreviated excerpt from their exposition might be the following. There were early contributions by Hölder (1901) and then solutions to restricted version of the question by Aczel (1948) and Tamari (1949); in [5] in 1958, Clifford pointed out that the arguments and results are further refined and expanded by theorems of Hofmann, Aczel, Criagan and Pales, among others (see [7]). Also a proof, using generalized metric techniques is given by Barnhart in [3]. That proof, however, is restricted to the abelian case. Here we give a fairly short proof using only generalized metric techniques. We also show that the theorem still holds if ‘connected’ is replaced by ‘separable’. One might assume that would be a corollary of the theorem in [7] that ‘a totally ordered set can be embedded in R if and only if it contains a countable subset C such that, for any x < y there is a c ∈ C with x ≤ c ≤ y’ [with no mention of a semigroup]. The assumption that that hypothesis follows