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| Natura: | Preprint |
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2024
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| Accesso online: | https://arxiv.org/abs/2412.06998 |
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| _version_ | 1866913604137123840 |
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| author | Ball, R. N. Hager, A. W. |
| author_facet | Ball, R. N. Hager, A. W. |
| contents | In the category \(\mathbf{V}\) of unital archimedean vector lattices, four notions of uniform completeness obtain. In all cases completeness requires the convergence of uniformly Cauchy sequences; the completions are distinguished by the manner in which the convergence is regulated. Ordinary uniform convergence is regulated by the canonical unit \(1\). Inner relative uniform convergence, here termed iru-convergence, is regulated by an arbitrary positive element. Outer relative uniform convergence, here termed oru-convergence, is regulated by an arbitrary positive element of a vector lattice containing the given object as a sub-vector lattice. *-convergence is equivalent to ordinary uniform convergence on certain specified quotients of the vector lattice. In each case the complete objects form a full monoreflective subcategory of \(\mathbf{V}\), denoted respectively \(\mathbf{ucV}\), \(\mathbf{irucV}\), \(\mathbf{orucV}\), and \(\mathbf{*cV}\). In this article we provide a unified development of these completions by means of a novel pointfree variant of the classical Yosida adjunction. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_06998 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The four uniform completions of a unital archimedean vector lattice Ball, R. N. Hager, A. W. Functional Analysis 06D22, 06F20, 18F70, 46A40 In the category \(\mathbf{V}\) of unital archimedean vector lattices, four notions of uniform completeness obtain. In all cases completeness requires the convergence of uniformly Cauchy sequences; the completions are distinguished by the manner in which the convergence is regulated. Ordinary uniform convergence is regulated by the canonical unit \(1\). Inner relative uniform convergence, here termed iru-convergence, is regulated by an arbitrary positive element. Outer relative uniform convergence, here termed oru-convergence, is regulated by an arbitrary positive element of a vector lattice containing the given object as a sub-vector lattice. *-convergence is equivalent to ordinary uniform convergence on certain specified quotients of the vector lattice. In each case the complete objects form a full monoreflective subcategory of \(\mathbf{V}\), denoted respectively \(\mathbf{ucV}\), \(\mathbf{irucV}\), \(\mathbf{orucV}\), and \(\mathbf{*cV}\). In this article we provide a unified development of these completions by means of a novel pointfree variant of the classical Yosida adjunction. |
| title | The four uniform completions of a unital archimedean vector lattice |
| topic | Functional Analysis 06D22, 06F20, 18F70, 46A40 |
| url | https://arxiv.org/abs/2412.06998 |