Abstract
In this paper, we introduce (quasi-)consistent spaces and (quasi-)adjacent spaces to characterize convexity spaces. Firstly, we show that convexity spaces can be characterized by quasi-consistent spaces. They can be induced by each other. In particular, each convexity space can be quasi-consistentizable. Every quasi-consistency $\mathcal{U}$ can induce two hull operators and thus determine different convexities $\mathcal{C}^{\mathcal{U}}$ and $\mathcal{C}_{\mathcal{U}}$. And $\mathcal{C}^{\mathcal{U}}=\mathcal{C}_{\mathcal{U}}$ holds when $\mathcal{U}$ is a consistency. Secondly, we use quasi-adjacent spaces to characterize convexity spaces. Each convexity space can be quasi-adjacentizable. In both of characterizations of convexity, remotehood systems play an important role in inducing convexity. Finally, we show there exists a close relation between a quasi-consistency and a quasi-adjacency. Furthermore, there exists a one-to-one correspondence between a quasi-adjacency and a fully ordered quasi-consistency. And we deeply study the relationships among these structures.
Supporting Institution
National Natural Science Foundation of China
Project Number
No. 11871097, No. 12271036
References
- [1] N. Bourbaki, Topologie générale ch. I et II, Paris, 1940.
- [2] Y. Dong and F.-G. Shi, On weak convex MV-algebras, Comm. Algebra 51 (7), 2759–2778, 2023.
- [3] V.A. Efremovic, Infinitesimal spaces, Dokl. Akad. Nauk SSSR 76, 341–343, 1951 .
- [4] R. Engelking, General topology, Heldermann, Berlin, 1989.
- [5] P. Fletcher and W.F. Lindgren, Quasi-Uniform Spaces, Lecture Notes in Pure Appl. Math., vol. 77, Dekker, New York, 1982.
- [6] S.P. Franklin, Some results on order-convexity, Amer. Math Monthly 62, 1962.
- [7] S.A. Naimpally and B.D. Warrack, Proximity spaces, Cambridge Univ., Cambridge, 1970.
- [8] T. Rapcsak, Geodesic convexity nonlinear optimization, J. Option. Theory App. 69, 169–183, 1991.
- [9] Ju.M. Smirnov, On proximity spaces, Mat. Sb. (N.S.) 31 (73), 543–574, 1952.
- [10] M.L.J. Van de Vel, Theory of convex structures, North Holland, N.Y. 1993.
- [11] M.L.J. Van de Vel, Binary convexities and distributive lattices, Proc. London Math. Soc. 48, 1–33, 1984.
- [12] X. Wei and F.-G. Shi, Convexity-preserving properties of partial binary operations with respect to filter convex structures on effect algebras, Internat. J. Theoret. Phys. 61 (7), 2022.
- [13] X. Wei and F.-G. Shi, Interval convexity of scale effect algebras, Comm. Algebra 51 (7), 2877–2894, 2023.
- [14] A. Weil, Sur les groupes topologiques et les groupes mesurés, C. R. Acad. Paris, 202, 1936.
- [15] Y. Yue, W. Yao and W.K. Ho, Applications of Scott-closed sets in convex structures, Topol. Appl. 314, 108093, 2022.
Abstract
Project Number
No. 11871097, No. 12271036
References
- [1] N. Bourbaki, Topologie générale ch. I et II, Paris, 1940.
- [2] Y. Dong and F.-G. Shi, On weak convex MV-algebras, Comm. Algebra 51 (7), 2759–2778, 2023.
- [3] V.A. Efremovic, Infinitesimal spaces, Dokl. Akad. Nauk SSSR 76, 341–343, 1951 .
- [4] R. Engelking, General topology, Heldermann, Berlin, 1989.
- [5] P. Fletcher and W.F. Lindgren, Quasi-Uniform Spaces, Lecture Notes in Pure Appl. Math., vol. 77, Dekker, New York, 1982.
- [6] S.P. Franklin, Some results on order-convexity, Amer. Math Monthly 62, 1962.
- [7] S.A. Naimpally and B.D. Warrack, Proximity spaces, Cambridge Univ., Cambridge, 1970.
- [8] T. Rapcsak, Geodesic convexity nonlinear optimization, J. Option. Theory App. 69, 169–183, 1991.
- [9] Ju.M. Smirnov, On proximity spaces, Mat. Sb. (N.S.) 31 (73), 543–574, 1952.
- [10] M.L.J. Van de Vel, Theory of convex structures, North Holland, N.Y. 1993.
- [11] M.L.J. Van de Vel, Binary convexities and distributive lattices, Proc. London Math. Soc. 48, 1–33, 1984.
- [12] X. Wei and F.-G. Shi, Convexity-preserving properties of partial binary operations with respect to filter convex structures on effect algebras, Internat. J. Theoret. Phys. 61 (7), 2022.
- [13] X. Wei and F.-G. Shi, Interval convexity of scale effect algebras, Comm. Algebra 51 (7), 2877–2894, 2023.
- [14] A. Weil, Sur les groupes topologiques et les groupes mesurés, C. R. Acad. Paris, 202, 1936.
- [15] Y. Yue, W. Yao and W.K. Ho, Applications of Scott-closed sets in convex structures, Topol. Appl. 314, 108093, 2022.
Details
| Primary Language | English |
|---|---|
| Subjects | Topology, Pure Mathematics (Other) |
| Journal Section | Mathematics |
| Authors | |
| Project Number | No. 11871097, No. 12271036 |
| Early Pub Date | April 14, 2024 |
| Publication Date | February 28, 2025 |
| Published in Issue | Year 2025 |