α -cut-complete Boolean algebras TeX Export

@article{citeulike:1421837, abstract = {Let A be a Boolean algebra, and \$ \alpha \$ an infinite cardinal number or the symbol \$ \infty \$. An \$ \alpha \$-cut in A is an ordered pair (F,H) of subsets of A, each of power \$ \le \alpha \$, with \$ F \le H \$ elementwise, with 0 as the meet of differences \$ h - f (h \in H, f \in F) \$. A is called \$ \alpha \$-cut-complete if for each \$ \alpha \$-cut (F,H) there is \$ a \in A \$ with \$ F \le a \le H \$ elementwise. We describe the simply-constructed \$ \alpha \$-cut-completion \$ A^\alpha \$, show that \$ \alpha \$-cut-completeness solves a natural \$ \alpha \$-injectivity problem, determine when \$ A^\alpha \$ is the \$ \alpha \$-completion, or the completion, and interpret most of that topologically in Stone spaces. Oddly, these considerations seem novel in Boolean algebras, while for lattice-ordered groups and vector lattices, and dually for topological spaces, the analogous theory, especially for \$ \alpha = \omega\_1 \$, has received considerable study.}, author = {Hager, A. W.}, citeulike-article-id = {1421837}, citeulike-linkout-0 = {http://dx.doi.org/10.1007/s000120050067}, doi = {10.1007/s000120050067}, journal = {Algebra Universalis}, keywords = {test}, month = {July}, number = {1}, pages = {57--70}, posted-at = {2007-06-29 09:51:42}, priority = {1}, title = {\alpha -cut-complete Boolean algebras}, url = {http://dx.doi.org/10.1007/s000120050067}, volume = {39}, year = {1998} }

TY - JOUR ID - citeulike:1421837 L3 - citeulike-article-id:1421837 N2 - Let A be a Boolean algebra, and $ \alpha $ an infinite cardinal number or the symbol $ \infty $. An $ \alpha $-cut in A is an ordered pair (F,H) of subsets of A, each of power $ \le \alpha $, with $ F \le H $ elementwise, with 0 as the meet of differences $ h - f (h \in H, f \in F) $. A is called $ \alpha $-cut-complete if for each $ \alpha $-cut (F,H) there is $ a \in A $ with $ F \le a \le H $ elementwise. We describe the simply-constructed $ \alpha $-cut-completion $ A^\alpha $, show that $ \alpha $-cut-completeness solves a natural $ \alpha $-injectivity problem, determine when $ A^\alpha $ is the $ \alpha $-completion, or the completion, and interpret most of that topologically in Stone spaces. Oddly, these considerations seem novel in Boolean algebras, while for lattice-ordered groups and vector lattices, and dually for topological spaces, the analogous theory, especially for $ \alpha = \omega_1 $, has received considerable study. IS - 1 JF - Algebra Universalis EP - 70 TI - \alpha -cut-complete Boolean algebras VL - 39 SP - 57 KW - test AU - Hager, AW PY - 1998/07/02/ UR - http://dx.doi.org/10.1007/s000120050067 ER -