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篇目详细内容 |
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【篇名】 |
Weakly distributive domains (II) |
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【刊名】 |
Frontiers of Computer Science in China |
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【刊名缩写】 |
Front. Comput. Sci. China |
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【ISSN】 |
1673-7350 |
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【EISSN】 |
1673-7466 |
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【DOI】 |
10.1007/s11704-007-0036-x |
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【出版社】 |
Higher Education Press and Springer-Verlag |
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【出版年】 |
2007 |
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【卷期】 |
1
卷4期 |
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【页码】 |
373-384
页,共
12
页 |
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【作者】 |
JIANG Ying;
ZHANG Guo - Qiang;
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【关键词】 |
weakly distributive domain; stable bifinite domain; meet-cpo; stable function; cartesian closed category |
【摘要】 |
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In our previous work (Inform. and Comput., 2005, 202: 87?103), we have shown that for any ω-algebraic meet-cpo D, if all higher-order stable function spaces built from D are ω-algebraic, then D is finitary. This accomplishes the first of a possible, two-step process in solving the problem raised (LNCS, 1991, 530: 16?33; Domains and lambda-calculi, Cambridge Univ. Press, 1998) whether the category of stable bifinite domains of Amadio-Droste-G?bel (LNCS, 1991, 530: 16?33; Theor. Comput. Sci., 1993, 111: 89?101) is the largest cartesian closed full sub-category within the category of ω-algebraic meet-cpos with stable functions. This paper presents the results of the second step, which is to show that for any ω - algebraic meet-cpo D satisfying axioms M and I to be contained in a cartesian closed full sub-category using ω - algebraic meet-cpos with stable functions, it must not violate MI∞. We introduce a new class of domains called weakly distributive domains and show that for these domains to be in a cartesian closed category using ω-algebraic meet-cpos, property MI∞ must not be violated. Further, we demonstrate that principally distributive domains (those for which each principle ideal is distributive) form a proper subclass of weakly distributive domains, and Birkhoff’s M3 and N5 (Introduction to Lattices and order, Cambridge Univ. Press, 2002) are weakly distributive (but non-distributive). Then, we establish characterization results for weakly distributive domains. We also introduce the notion of meet-generators in constructing stable functions and show that if an ω-algebraic meet-cpo D contains an infinite number of meet-generators, then [D→D] fails I. However, the original problem of Amadio and Curien remains open. |
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