Quantale-Valued Gauge Groups and Approach Convergence Transformation Groups
Topology + Algebra and Analysis
32nd Summer Conference on Topology and Its Applications
E. Colebunders, et al., introduced a category C , consisting of objects all triple (X, S, S, δ), where X 2 | CAP|, an object in the category of Lowen-approach spaces , S 2 |CAG|, an object in the category of approach groups , and δ : X ⇥ S → X, a contraction mapping. Actually, in , the authors brought to light a concept of approach convergence transformation monoids without explicit mention. On the other hand, following the idea of probabilistic convergence group  (see also ), we introduced a category of probabilistic convergence transformation groups, PCONVTG . Our motive here is to demonstrate the link between these two categories. In so doing and, failing to provide a direct link between these two, apparently different approaches, we consider a value quantale V in the line of [6,7] (see also , with opposite order), and propose a notion of quantale-valued gauge group (en route to a category V-CONVTG) - a notion closely related to quantale-valued metric group vis-`a-vis quantale-valued convergence group. The advantage that we have using V-CONVTG is, it provides a global framework, where C, like many others existing categories of similar nature, serve examples whenever appropriate quantales are considered.
 T. M. G. Ahsanullah and G. Jäger, Probabilistic uniformizability and probabilistic metrizability of probabilistic convergence groups, to appear in Mathematica Slovaca.
 T. M. G. Ahsanullah and G. Jäger, Probabilistic convergence transformation groups, submitted for publication, preprint, 2017.
 E. Colebunders, H. Boustique, P. Mikusi´nski, G. Richardson, Convergence approach spaces: Actions, Applied Categorical Structures 24(2016), 147–161.
 R. C. Flagg, Quantales and continuity spaces, Algebra Univers. 37(1997), 257–276.
 G. J¨ager, A convergence theory for probabilistic metric spaces, Quaest. Math. 3(2015), 587-599.
 G. J¨ager and W. Yao, Quantale-valued gauge spaces, to appear in Iranian Journal of Fuzzy Systems.
 H. Lai and W. Tholen, Quantale-valued approach spaces via closure and convergence, arXiv:1604.08813.
 R. Lowen, Approach Spaces: The Missing Link in the Topology-Uniformity-Metric Triad, Clarendon Press, Oxford, 1997. Index Analysis, Springer, 2016.
 R. Lowen and B. Windels, Approach groups, Rocky Mountain J. Math. 30(2000), 1057–1073.
Copyright © 2017, the Author
Ahsanullah, T.M.G., "Quantale-Valued Gauge Groups and Approach Convergence Transformation Groups" (2017). Summer Conference on Topology and Its Applications. 62.