Completeness results for metrized rings and lattices | ||
| Categories and General Algebraic Structures with Applications | ||
| مقاله 10، دوره 11، Special Issue Dedicated to Prof. George A. Grätzer، مهر 2019، صفحه 149-168 اصل مقاله (739.35 K) | ||
| نوع مقاله: Research Paper | ||
| شناسه دیجیتال (DOI): 10.29252/cgasa.11.1.149 | ||
| نویسنده | ||
| George M. Bergman | ||
| University of California, Berkeley | ||
| چکیده | ||
| The Boolean ring $B$ of measurable subsets of the unit interval, modulo sets of measure zero, has proper radical ideals (for example, $\{0\})$ that are closed under the natural metric, but has no prime ideal 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 same 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; that is, $L$ 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. | ||
| کلیدواژهها | ||
| Complete topological ring without closed prime ideals؛ measurable sets modulo sets of measure zero؛ lattice complete under a metric | ||
| مراجع | ||
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[1] Cohn, P. M., "Basic Algebra. Groups, Rings and Fields", Springer, 2003. [2] Fremlin, D. H., "Measure Theory. Vol. 3. Measure Algebras", corrected second printing of the 2002 original. Torres Fremlin, 2004. [3] Halmos, P. R., "Measure Theory", D. Van Nostrand Company, 1950. [4] Lang, S., "Real and Functional Analysis. Third edition", Graduate Texts in Mathematics 142, Springer, 1993. [5] Mennucci, A., The metric space of (measurable) sets, and Carathéodory’s theorem, (2013), 3 Pages, readable at http://dida.sns.it/dida2/cl/13-14/folde2/pdf1. | ||
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