Compactness of the space of left orders. (English) Zbl 1129.57024
A magma is a set with a binary operation. A left order on a magma is an order of the magma elements which is invariant under the magma operation. The set of all such left orders was topologized in [A. S. Sikora, Bull. Lond. Math. Soc. 36, No. 4, 519–526 (2004; Zbl 1057.06006)] in the case of semigroups.
In this paper, the authors extend the previous work to arbitrary magmas, topologizing the set of left orders of a magma by choosing a subbasis whose elements are sets of all left orders containing a specified pair of distinct elements. They show that the resulting topological space is zero-dimensional, compact, and, if countable, metrizable. Results are obtained about right- and bi-orderings of quandles (a type of non-associative magma) and a connection is made with a theorem of Conrad. Finally, open questions and examples are collected.
In this paper, the authors extend the previous work to arbitrary magmas, topologizing the set of left orders of a magma by choosing a subbasis whose elements are sets of all left orders containing a specified pair of distinct elements. They show that the resulting topological space is zero-dimensional, compact, and, if countable, metrizable. Results are obtained about right- and bi-orderings of quandles (a type of non-associative magma) and a connection is made with a theorem of Conrad. Finally, open questions and examples are collected.
Reviewer: Sam Nelson (Rancho Cucamonga)
MSC:
| 57M99 | General low-dimensional topology |
| 17A99 | General nonassociative rings |
| 03G05 | Logical aspects of Boolean algebras |
| 54D30 | Compactness |
Citations:
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