Summer Conference on Topology and Its Applications
 

Title

Quantale-Valued Gauge Groups and Approach Convergence Transformation Groups

Document Type

Topology + Algebra and Analysis

Publication Date

6-2017

Publication Source

32nd Summer Conference on Topology and Its Applications

Abstract

E. Colebunders, et al., introduced a category C [3], consisting of objects all triple (X, S, S, δ), where X 2 | CAP|, an object in the category of Lowen-approach spaces [8], S 2 |CAG|, an object in the category of approach groups [9], and δ : X ⇥ S → X, a contraction mapping. Actually, in [3], 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 [1] (see also [5]), we introduced a category of probabilistic convergence transformation groups, PCONVTG [2]. 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 [4], 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.

References

[1] T. M. G. Ahsanullah and G. Jäger, Probabilistic uniformizability and probabilistic metrizability of probabilistic convergence groups, to appear in Mathematica Slovaca.

[2] T. M. G. Ahsanullah and G. Jäger, Probabilistic convergence transformation groups, submitted for publication, preprint, 2017.

[3] E. Colebunders, H. Boustique, P. Mikusi´nski, G. Richardson, Convergence approach spaces: Actions, Applied Categorical Structures 24(2016), 147–161.

[4] R. C. Flagg, Quantales and continuity spaces, Algebra Univers. 37(1997), 257–276.

[5] G. J¨ager, A convergence theory for probabilistic metric spaces, Quaest. Math. 3(2015), 587-599.

[6] G. J¨ager and W. Yao, Quantale-valued gauge spaces, to appear in Iranian Journal of Fuzzy Systems.

[7] H. Lai and W. Tholen, Quantale-valued approach spaces via closure and convergence, arXiv:1604.08813.

[8] R. Lowen, Approach Spaces: The Missing Link in the Topology-Uniformity-Metric Triad, Clarendon Press, Oxford, 1997. Index Analysis, Springer, 2016.

[9] R. Lowen and B. Windels, Approach groups, Rocky Mountain J. Math. 30(2000), 1057–1073.

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