Completeness results for metrized rings and lattices
| Autor: | George M. Bergman |
|---|---|
| Rok vydání: | 2018 |
| Předmět: |
Modulo
Prime ideal 54E50 13J10 06B35 (Primary) 13A15 54E50 (Secondary) Cauchy sequence Null set Combinatorics Lattice (order) QA1-939 FOS: Mathematics Mathematics - Combinatorics Discrete Mathematics and Combinatorics 06B35 math.CO math.RA Mathematics 13A15 lattice complete under a metric Applied Mathematics measurable sets modulo sets of measure zero 13J10 Boolean ring Mathematics - Rings and Algebras Computational Mathematics Metric space Rings and Algebras (math.RA) complete topological ring without closed prime ideals Combinatorics (math.CO) 06B35 (Primary) Analysis |
| Zdroj: | Categories and General Algebraic Structures with Applications, vol 11, iss SpecialIssue Categories and General Algebraic Structures with Applications, Vol 11, Iss Special Issue Dedicated to Prof. George A. Gratzer, Pp 149-168 (2019) |
| DOI: | 10.48550/arxiv.1808.04455 |
| Popis: | The Boolean ring $B$ of measurable subsets of the unit interval, modulo sets of measure zero, has proper radical ideals (e.g., $\{0\})$ that are closed under the natural metric, but has no prime ideals closed under that metric; hence closed radical ideals are not, in general, intersections of closed prime ideals. Moreover, $B$ is known to be complete in its metric. Together, these facts answer a question posed by J.Gleason. From this example, rings of arbitrary characteristic with the corresponding properties are obtained. The result that $B$ is complete in its metric is generalized to show that if $L$ is a lattice given with a metric satisfying identically either the inequality $d(x\vee y,\,x\vee z)\leq d(y,z)$ or the inequality $d(x\wedge y,\,x\wedge z)\leq d(y,z),$ and if in $L$ every increasing Cauchy sequence converges and every decreasing Cauchy sequence converges, then every Cauchy sequence in $L$ converges; i.e., $L$ is complete as a metric space. We show by example that if the above inequalities are replaced by the weaker conditions $d(x,\,x\vee y)\leq d(x,y),$ respectively $d(x,\,x\wedge y)\leq d(x,y),$ the completeness conclusion can fail. We end with two open questions. Comment: Lots of local rewriting of the previous version; several embarrassing misstatements (e.g., "primary" instad of "radical" in the abstract) corrected |
| Databáze: | OpenAIRE |
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