A topological structure on certain initial algebras. (English) Zbl 1326.06014
Summary: There is a well-known natural topology on the set of compatible total orders on a group, and recent results of A. Clay [Monatsh. Math. 167, No. 3-4, 417-430 (2012; Zbl 1257.06008)], M. A. Dabkowska et al. [J. Knot Theory Ramifications 16, No. 3, 257-266 (2007; Zbl 1129.57024)], and A. S. Sikora [Bull. Lond. Math. Soc. 36, No. 4, 519-526 (2004; Zbl 1057.06006)] have characterized this topology for certain groups. We consider a similar topology on the set of distinct monomial algebras in polynomial and Laurent polynomial rings. We study the latter topological structure for monomial algebras that come from rings of multiplicative invariants and show that they are either finite discrete spaces or homeomorphic to the Cantor set.
MSC:
| 06F15 | Ordered groups |
| 13P10 | Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases) |
| 54H13 | Topological fields, rings, etc. (topological aspects) |
Keywords:
Cantor set; orderable groups; monomial algebras; initial algebras; multiplicative invariants; Laurent polynomial ringsReferences:
| [1] | Brouwer, L. E.J., Over de structuur der perfekte puntverzamelingen, Amst. Akad. Versl., 18, 833-842 (1910) · JFM 41.0103.03 |
| [2] | Clay, A., Free lattice-ordered groups and the space of left orderings, Monatshefte Math., 167, 417-430 (2012) · Zbl 1257.06008 |
| [3] | Cox, D.; Little, J.; Shank, H., Toric Varieties, Grad. Stud. Math., vol. 124 (2011), American Mathematical Society: American Mathematical Society Providence, RI · Zbl 1223.14001 |
| [4] | Dabkowska, M. A.; Dabkowski, M. K., Compactness of the space of left orders, J. Knot Theory Ramif., 16, 3, 257-266 (2007) · Zbl 1129.57024 |
| [5] | Kapur, D.; Madlener, K., A completion procedure for computing a canonical bases for a \(k\)-subalgebra, (Kaltofen, E.; Watt, S., Proceedings of Computers and Mathematics, vol. 89 (1989), MIT: MIT Cambridge, Mass), 1-11 · Zbl 0692.13001 |
| [6] | Kuroda, S., The infiniteness of the SAGBI bases for certain invariant rings, Osaka J. Math., 39, 665-680 (2002) · Zbl 1045.13012 |
| [7] | Linnell, P., The space of left orders of a group is either finite or uncountable, Bull. Lond. Math. Soc., 43, 200-2002 (2011) · Zbl 1215.06009 |
| [8] | Lorenz, M., Multiplicative invariant theory, (Invariant Theory and Algebraic Transformation Groups. Invariant Theory and Algebraic Transformation Groups, Encyclopaedia Math. Sci., vol. 135 (2005), Springer-Verlag: Springer-Verlag Berlin Heidelberg) · Zbl 1078.13003 |
| [9] | Reichstein, Z., SAGBI bases in rings of multiplicative invariants, Comment. Math. Helv., 78, 1, 185-202 (2003) · Zbl 1043.13008 |
| [10] | Robbiano, L., Term orders on the polynomial ring, (Proceedings of the EUROCAL 85. Proceedings of the EUROCAL 85, Lect. Notes Comput. Sci., vol. 204 (1985)), 513-517 · Zbl 0584.13016 |
| [11] | Robbiano, L.; Sweedler, M., Subalgebra bases, (Commutative Algebra. Commutative Algebra, Salvador, 1988. Commutative Algebra. Commutative Algebra, Salvador, 1988, Lect. Notes Math., vol. 1430 (1990), Springer: Springer Berlin), 61-87 · Zbl 0725.13013 |
| [12] | Sikora, A., Topology on the space of orderings of groups, Bull. Lond. Math. Soc., 36, 519-526 (2004) · Zbl 1057.06006 |
| [13] | Tesemma, M., On initial algebra of multiplicative invariants, J. Algebra, 320, 3851-3865 (2008) · Zbl 1154.13004 |
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