| Title:
|
Multi-argument specialization semilattices (English) |
| Author:
|
Lipparini, Paolo |
| Language:
|
English |
| Journal:
|
Mathematica Bohemica |
| ISSN:
|
0011-4642 |
| ISSN:
|
0862-7959 (print) |
| ISSN:
|
2464-7136 (online) |
| Volume:
|
150 |
| Issue:
|
4 |
| Year:
|
2025 |
| Pages:
|
537-559 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
If $X$ is a closure space with closure $K$, we consider the semilattice $(\mathcal P(X), \cup )$ endowed with a further relation $ x \sqsubseteq \{ y_1, y_2, \dots , y_n\} $ between elements of $\mathcal P(X)$ and finite subsets of $\mathcal P(X)$, whose interpretation is $x \subseteq Ky_1 \cup Ky_2 \cup \dots \cup Ky_n $. \endgraf We present axioms for such multi-argument specialization semilattices and show that this list of axioms is sound and complete for substructures of closure spaces, namely, a model satisfies the axioms if and only if it can be embedded into the structure associated to a closure space as in the previous sentence. As a main tool for the proof, we provide a canonical embedding of a multi-argument specialization semilattice into (the structure associated to) a closure semilattice. (English) |
| Keyword:
|
multi-argument specialization semilattice |
| Keyword:
|
closure semilattice |
| Keyword:
|
closure space |
| Keyword:
|
universal extension |
| MSC:
|
06A12 |
| MSC:
|
06A15 |
| MSC:
|
06F99 |
| MSC:
|
54A05 |
| DOI:
|
10.21136/MB.2025.0082-24 |
| . |
| Date available:
|
2025-11-07T18:37:41Z |
| Last updated:
|
2025-11-16 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/153161 |
| . |
| Reference:
|
[1] Blass, A.: Combinatorial cardinal characteristics of the continuum.Handbook of Set Theory Springer, Dordrecht (2010), 395-489. Zbl 1198.03058, MR 2768685, 10.1007/978-1-4020-5764-9_7 |
| Reference:
|
[2] Blyth, T. S.: Lattices and Ordered Algebraic Structures.Universitext. Springer, London (2005). Zbl 1073.06001, MR 2126425, 10.1007/b139095 |
| Reference:
|
[3] Caspard, N., Leclerc, B., Monjardet, B.: Finite Ordered Sets: Concepts, Results and Uses.Encyclopedia of Mathematics and its Applications 144. Cambridge University Press, Cambridge (2012). Zbl 1238.06001, MR 2919901, 10.1017/CBO9781139005135 |
| Reference:
|
[4] Čech, E.: Topological Spaces.Publishing House of the Czechoslovak Academy of Sciences, Prague; John Wiley & Sons, London (1966). Zbl 0141.39401, MR 0211373 |
| Reference:
|
[5] Chajda, I., Halaš, R., Kühr, J.: Semilattice Structures.Research and Exposition in Mathematics 30. Heldermann Verlag, Lemgo (2007). Zbl 1117.06001, MR 2326262 |
| Reference:
|
[6] Császár, Á.: Generalized topology, generalized continuity.Acta Math. Hung. 96 (2002), 351-357. Zbl 1006.54003, MR 1922680, 10.1023/A:1019713018007 |
| Reference:
|
[7] Erné, M.: Closure.Beyond Topology Contemporary Mathematics 486. AMS, Providence (2009), 163-238. Zbl 1192.54001, MR 2521945, 10.1090/conm/486 |
| Reference:
|
[8] Gierz, G., Hofmann, K., Keimel, K., Lawson, J., Mislove, M., Scott, D. S.: Continuous Lattices and Domains.Encyclopedia of Mathematics and its Applications 93. Cambridge University Press, Cambridge (2003). Zbl 1088.06001, MR 1975381, 10.1017/CBO9780511542725 |
| Reference:
|
[9] Harzheim, E.: Ordered Sets.Advances in Mathematics 7. Springer, New York (2005). Zbl 1072.06001, MR 2127991, 10.1007/b104891 |
| Reference:
|
[10] Hodges, W.: Model Theory.Encyclopedia of Mathematics and its Applications 42. Cambridge University Press, Cambridge (1993). Zbl 0789.03031, MR 1221741, 10.1017/CBO9780511551574 |
| Reference:
|
[11] Jansana, R.: Propositional consequence relations and algebraic logic.The Stanford Encyclopedia of Philosophy (Spring 2011 Edition) Stanford University, Stanford (2011), Available at \brokenlink {https://plato.stanford.edu/archives/spr2011/entries/{consequence-algebraic/}}\kern0pt. |
| Reference:
|
[12] Kronheimer, E. H., Penrose, R.: On the structure of causal spaces.Proc. Camb. Philos. Soc. 63 (1967), 481-501. Zbl 0148.46502, MR 0208982, 10.1017/S030500410004144X |
| Reference:
|
[13] Lipparini, P.: Universal extensions of specialization semilattices.Categ. Gen. Algebr. Struct. Appl. 17 (2022), 101-116. Zbl 07626867, MR 4475652, 10.52547/cgasa.2022.102467 |
| Reference:
|
[14] Lipparini, P.: Preservation of superamalgamation by expansions.Available at https://arxiv.org/abs/2203.10570v3 (2023), 42 pages. 10.48550/arXiv.2203.10570 |
| Reference:
|
[15] Lipparini, P.: Universal specialization semilattices.Quaest. Math. 46 (2023), 2163-2176. Zbl 07745971, MR 4644164, 10.2989/16073606.2022.2126805 |
| Reference:
|
[16] Lipparini, P.: Semilattices with a congruence.Available at https://arxiv.org/abs/2407.08446 (2024), 9 pages. 10.48550/arXiv.2407.08446 |
| Reference:
|
[17] Lipparini, P.: A model theory of topology.(to appear) in Stud. Logica. 10.1007/s11225-024-10107-3 |
| Reference:
|
[18] McKinsey, J. C. C., Tarski, A.: The algebra of topology.Ann. Math. (2) 45 (1944), 141-191. Zbl 0060.06206, MR 0009842, 10.2307/1969080 |
| Reference:
|
[19] Mynard, F., Pearl, E., eds.: Beyond Topology.Contemporary Mathematics 486. AMS, Providence (2009). Zbl 1165.54001, MR 2555999, 10.1090/conm/486 |
| Reference:
|
[20] Nagata, J.: Modern General Topology.North-Holland Mathematical Library 33. North-Holland, Amsterdam (1985). Zbl 0598.54001, MR 0831659 |
| Reference:
|
[21] Ranzato, F.: Closures on CPOs form complete lattices.Inf. Comput. 152 (1999), 236-249. Zbl 1004.06007, MR 1707895, 10.1006/inco.1999.2801 |
| Reference:
|
[22] Rump, W.: Multi-posets in algebraic logic, group theory, and non-commutative topology.Ann. Pure Appl. Logic 167 (2016), 1139-1160. Zbl 1377.08002, MR 3541004, 10.1016/j.apal.2016.05.001 |
| Reference:
|
[23] Scowcroft, P.: Existentially closed closure algebras.Notre Dame J. Formal Logic 61 (2020), 623-661. Zbl 1486.03067, MR 4200354, 10.1215/00294527-2020-0026 |
| Reference:
|
[24] Servi, M.: Algebre di Fréchet: Una classe di algebre booleane con operatore.Rend. Circ. Mat. Palermo, II. Ser. 14 (1965), 335-366 Italian. Zbl 0158.40902, MR 0210059, 10.1007/BF02844036 |
| Reference:
|
[25] Vickers, S.: Topology Via Logic.Cambridge Tracts in Theoretical Computer Science 5. Cambridge University Press, Cambridge (1989). Zbl 0668.54001, MR 1002193 |
| . |
