Exactness of Proximal Group Homomorphisms
Abstract
This research introduces groups in proximity spaces which endowed with a proximity relation. Two penultimate choices for such relations are the Efremovic (EF) proximity relation and its extension, namely, the descriptive EF-proximity relation. There is a strong relationship between sets (groups) and set (group) descriptions. Therefore, in this paper we consider this relationship via exactness of descriptive homomorphisms between ordinary descriptive groups and meta-descriptive groups. The definition of a short exact sequence of descriptive homomorphisms is given. Then, results were obtained giving the relationships between the two short exact sequences.
Keywords
Proximal group, EF-proximity space, Descriptive EF-proximity space, Descriptive homomorphism.
Thanks
The author wish to thank the anonymous reviewers for their valuable suggestions.
References
- [1] Peters, J.F., İnan, E., Öztürk, M.A., Spatial and descriptive isometries in proximity spaces, Gen. Math. Notes 21(2) (2014) 1-10.
- [2] Peters, J.F., İnan, E., Öztürk, M.A., Monoids in proximal Banach spaces, Int. J. Algebra 8(18) (2014) 869-872.
- [3] Peters, J.F., Naimpally, S., Applications of near sets, Notes of the Amer. Math. Soc. 59(4) (2012) 536-542.
- [4] Peters, J.F., Near sets: An introduction, Math. Comput. Sci. 7(1) (2013) 3-9.
- [5] Efremovic, V.A., The geometry of proximity I (in Russian), Mat. Sbornik N. S. 31(73) (1952) 189-200.
- [6] Kasch, F., Modules and Rings, Academic Press Inc. Ltd., London, 1982.
- [7] Kovar, M.M., A new causal topology and why the universe is co-compact, arXive:1112.0817 [math-ph] (2011) 1-15.
- [8] Peters, J.F., Local near sets: Pattern discovery in proximity spaces, Math. Comput. Sci. 7(1) (2013) 87-106.
- [9] Cech, E., Topological Spaces, John Wiley & Sons Ltd., London, 1966.
- [10] Peters, J.F., Öztürk, M.A., Uçkun, M., Exactness of Proximal Groupoid Homomorphisms, Adıyaman University Journal of Science 5(1) (2015) 1-13.