Continuous prequantale models of \(T_1\) topological semigroups. (English) Zbl 07513475
Jung, Achim (ed.) et al., Proceedings of the 8th international symposium on domain theory and its applications, ISDT 2019, Yangzhou, China, June 14–17, 2019. Amsterdam: Elsevier. Electron. Notes Theor. Comput. Sci. 345, 99-111 (2019).
Summary: In this paper, we show that every \(T_1\) topological semigroup satisfying condition \((\Delta)\) can be embedded into a topological semigroup \((D, \sigma, \odot)\), where \((D, \sqsubseteq)\) is a domain. Furthermore, by considering the maximal point topological semigroup of a continuous prequantale, it is proven that every \(T_1\) topological semigroup satisfying condition \((\Delta)\) has a continuous prequantale model, which may not be bounded complete.
For the entire collection see [Zbl 1420.68011].
For the entire collection see [Zbl 1420.68011].
MSC:
| 68Q55 | Semantics in the theory of computing |
| 06B35 | Continuous lattices and posets, applications |
| 06F07 | Quantales |
| 22A25 | Representations of general topological groups and semigroups |
Keywords:
topological semigroup; prequantale; stable ordered semigroup; Scott topology; prequantale modelReferences:
| [1] | Ali-Akbari, M.; Honari, B.; Pourmahdian, M., Any \(T_1\) space has a continuous poset model, Topology and its Applications, 156, 2240-2245 (2009) · Zbl 1173.54010 |
| [2] | Edalat, A.; Heckmann, R., A computational model for metric spaces, Theoretical Computer Science, 193, 53-73 (1998) · Zbl 1011.54026 |
| [3] | Erné, M., Algebraic models for \(T_1\)-spaces, Topology and its Applications, 158, 945-962 (2011) · Zbl 1216.54002 |
| [4] | Kamimura, T.; Tang, A., Total objects of domains, Theoretical Computer Science, 34, 275-288 (1984) · Zbl 0551.68047 |
| [5] | Kopperman, R.; Künzi, H.-P. A.; Waszkiewicz, P., Bounded complete models of topological spaces, Topology and its Applications, 139, 285-297 (2004) · Zbl 1054.06004 |
| [6] | Lawson, J. D., The round ideal completion via sobrification, Topology Proceedings, 22, 261-274 (1997) · Zbl 0948.06004 |
| [7] | Lawson, J. D., Spaces of maximal points, Mathematical Structures in Computer Science, 7, 543-555 (1997) · Zbl 0985.54025 |
| [8] | Liang, J. H.; Keimel, K., Order environments of topological spaces, Acta Mathematica Sinica, English Series, 20, 943-948 (2004) · Zbl 1072.54021 |
| [9] | Martin, K., Domain theoretic models of topological spaces, Electronic Notes in Theoretical Computer Science, 13, 173-181 (1998) |
| [10] | Martin, K., Ideal models of spaces, Theoretical Computer Science, 305, 277-297 (2003) · Zbl 1044.54005 |
| [11] | Martin, K., The regular spaces with countably based models, Theoretical Computer Science, 305, 299-310 (2003) · Zbl 1053.54037 |
| [12] | Scott, D. S., Continuous lattices, (Lawvere, F. W., Toposes, Algebraic Geometry and Logic. Toposes, Algebraic Geometry and Logic, Lecture Notes in Mathematics, vol. 274 (1972)), 97-136 · Zbl 0239.54006 |
| [13] | Zhao, B.; Xia, C.; Wang, K., Topological semigroups and their prequantale models, Filomat, 31, 6205-6210 (2017) · Zbl 1499.22008 |
| [14] | Zhao, D., Poset models of topological spaces, (Proceeding of International Conference on Quantitative Logic and Quantification of Software (2009), Global-link Publisher), 229-238 |
| [15] | Zhao, D.; Xi, X., Directed complete poset models of \(T_1\) spaces, Mathematical Proceedings of the Cambridge Philosophical Society, 164, 125-134 (2018) · Zbl 1469.06012 |
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.
