Topology on the Spaces of Orderings of Groups and Semi-goups

We introduce a natural topology on the space of left orderings of an arbitrary semi-group G. We prove that this space is always compact and that for free abelian groups it is homeomorphic to the Cantor set. We use our topological approach to provide a simple proof of the existence of universal Gröbner bases. 1. Orderings for semi-groups Given a semi-group G (ie. a set with an associative binary operation), a linear order, <, on G is a left ordering if a < b implies ca < cb, for any c. Similarly, a linear order , <, is a right ordering if a < b implies ac < bc, for any c ∈ G. The sets of all left and right orderings of G are denoted by LO(G) and RO(G) respectively. If G is a group then there is a 1-1 correspondence between these two sets which associates with any left ordering, <l, a right ordering, <r, such that a <r b if and only if b −1 <l a −1. For more about ordering of groups see [KK, MR]. Let Ua,b ⊂ LO(G) denote the set of all left orderings, <, for which a < b. We can put a topology on LO(G) in one of the following two ways. Definition 1.1. LO(G) has the smallest topology for which all the sets Ua,b are open. Any open set in this topology is a union of sets of the form Ua1,b1 ∩ ... ∩ Uan,bn . Definition 1.2. Let G0 ⊂ G1 ⊂ G2... ⊂ G be an arbitrary complete filtration of G by its subsets. (A filtration is complete if ⋃ i Gi = G). For <1, <2∈ LO(G) we define ρ(<1, <2) to be 1 2 , where r is the largest number with the property that <1 and <2 coincide when restricted to Gr; We put ρ(<1, <2) = 0 if such r does not exist (r = ∞). From now on we will consider only countable semi-groups G, and only such filtrations which are composed of finite subsets of G. Proposition 1.3. ρ is a metric on LO(G) and the topology on LO(G) induced by that metric coincides with the topology introduced in Definition 1.1. In particular, it does not depend on the choice of a filtration of G. Proof. It is easy to check that ρ is a metric. Hence, the proposition follows from the following two statements: (1) any open ball B(<0, 1/2 ) (with respect to the metric ρ) is open in the