×

A quantitative computational model for complete partial metric spaces via formal balls. (English) Zbl 1172.06003

The authors present a computational model for partial metric spaces that generalizes and unifies existing work in the literature.
Given a partial metric space \((X,p)\), they use \(({\mathbb B}X,\sqsubseteq_{d_p})\) to denote the poset of formal balls of the associated quasi-metric space \((X,d_p).\) They prove that a partial metric space \((X,p)\) is complete if and only if the poset \(({\mathbb B}X,\sqsubseteq_{d_p})\) is a domain. Moreover, \((X,p)\) is complete and sup-separable if and only if \(({\mathbb B}X,\sqsubseteq_{d_p})\) is an \(\omega\)-domain.
Furthermore, with the help of the concept of a partial quasi-metric, for any complete partial metric space \((X,p)\) they construct a Smyth complete quasi-metric \(q\) on \({\mathbb B}X\) that extends the quasi-metric \(d_p\) such that both the Scott topology and the partial order \(\sqsubseteq_{d_p}\) are induced by \(q.\)

MSC:

06B35 Continuous lattices and posets, applications
54E35 Metric spaces, metrizability
68Q55 Semantics in the theory of computing
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] DOI: 10.1016/S0895-7177(02)00100-0 · Zbl 1063.68057 · doi:10.1016/S0895-7177(02)00100-0
[2] DOI: 10.1016/S1571-0661(04)80975-7 · Zbl 1260.68222 · doi:10.1016/S1571-0661(04)80975-7
[3] Fletcher, Quasi-Uniform Spaces (1982)
[4] DOI: 10.1016/S1571-0661(05)80164-1 · doi:10.1016/S1571-0661(05)80164-1
[5] Smyth, Topology and Category Theory in Computer Science pp 207– (1991)
[6] Engelking, General Topology (1977)
[7] Smyth, Springer-Verlag Lecture Notes in Computer Science 298 pp 236– (1988) · doi:10.1007/3-540-19020-1_12
[8] DOI: 10.1016/S0304-3975(98)00288-6 · Zbl 0916.68045 · doi:10.1016/S0304-3975(98)00288-6
[9] DOI: 10.1016/j.tcs.2003.11.016 · Zbl 1052.54026 · doi:10.1016/j.tcs.2003.11.016
[10] DOI: 10.1016/S0304-3975(96)00243-5 · Zbl 1011.54026 · doi:10.1016/S0304-3975(96)00243-5
[11] DOI: 10.1016/S0304-3975(02)00705-3 · Zbl 1043.54011 · doi:10.1016/S0304-3975(02)00705-3
[12] DOI: 10.1006/inco.1995.1096 · Zbl 0834.58029 · doi:10.1006/inco.1995.1096
[13] DOI: 10.1016/S1571-0661(04)00029-5 · Zbl 0910.68135 · doi:10.1016/S1571-0661(04)00029-5
[14] DOI: 10.1016/0304-3975(95)00050-7 · Zbl 0872.28006 · doi:10.1016/0304-3975(95)00050-7
[15] DOI: 10.1016/S0166-8641(97)00224-1 · Zbl 0982.54029 · doi:10.1016/S0166-8641(97)00224-1
[16] DOI: 10.1016/j.topol.2005.01.023 · Zbl 1084.22002 · doi:10.1016/j.topol.2005.01.023
[17] DOI: 10.1016/S0166-8641(98)00102-3 · Zbl 0941.54028 · doi:10.1016/S0166-8641(98)00102-3
[18] DOI: 10.1007/s10474-005-0208-9 · Zbl 1092.54010 · doi:10.1007/s10474-005-0208-9
[19] DOI: 10.1006/jath.1999.3439 · Zbl 0980.41029 · doi:10.1006/jath.1999.3439
[20] Romaguera, Mathematische Nachrichten 157 pp 15– (1992)
[21] DOI: 10.1007/BF01301400 · Zbl 0472.54018 · doi:10.1007/BF01301400
[22] DOI: 10.1111/j.1749-6632.1996.tb49177.x · doi:10.1111/j.1749-6632.1996.tb49177.x
[23] DOI: 10.1007/BF02871458 · Zbl 1098.54027 · doi:10.1007/BF02871458
[24] DOI: 10.1111/j.1749-6632.1994.tb44144.x · Zbl 0911.54025 · doi:10.1111/j.1749-6632.1994.tb44144.x
[25] Martin, Springer-Verlag Lecture Notes in Computer Science 1853 pp 116– (2000) · doi:10.1007/3-540-45022-X_11
[26] DOI: 10.1016/S1571-0661(05)80221-X · doi:10.1016/S1571-0661(05)80221-X
[27] DOI: 10.1017/S0960129597002363 · Zbl 0985.54025 · doi:10.1017/S0960129597002363
[28] DOI: 10.1111/j.1749-6632.1994.tb44134.x · doi:10.1111/j.1749-6632.1994.tb44134.x
[29] DOI: 10.1016/S0304-3975(00)00335-2 · Zbl 1025.54014 · doi:10.1016/S0304-3975(00)00335-2
[30] DOI: 10.1016/j.tcs.2006.07.050 · Zbl 1109.54021 · doi:10.1016/j.tcs.2006.07.050
[31] Künzi, Handbook of the History of General Topology pp 853– (2001) · doi:10.1007/978-94-017-0470-0_3
[32] Künzi, Proc. Szekszárd Conf. pp 303– (1993)
[33] DOI: 10.1016/j.tcs.2006.05.037 · Zbl 1171.68542 · doi:10.1016/j.tcs.2006.05.037
[34] DOI: 10.1016/j.topol.2003.12.001 · Zbl 1054.06004 · doi:10.1016/j.topol.2003.12.001
[35] DOI: 10.1023/A:1008684018933 · Zbl 0993.54029 · doi:10.1023/A:1008684018933
[36] Gierz, Continuous Lattices and Domains (2003) · Zbl 1088.06001 · doi:10.1017/CBO9780511542725
[37] DOI: 10.1017/S0960129506005196 · Zbl 1103.06004 · doi:10.1017/S0960129506005196
[38] DOI: 10.1023/A:1023012924892 · Zbl 1030.06005 · doi:10.1023/A:1023012924892
[39] DOI: 10.1007/BF01874606 · Zbl 0845.54016 · doi:10.1007/BF01874606
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.