Quantale-valued convergence transformation groups
Considering a value quantale, we introduce a category of quantale-valued convergence transformation groups on quantale-valued convergence spaces. A wide variety of examples including probabilistic convergence transformation group, probabilistic metric transformation group, and convergence approach transformation group are presented, while interrelationships among quantale-valued convergence transformation groups, quantale-valued gauge transformation groups, approach system transformation groups and quantale-valued metric transformation groups are projected. We depict a global view of our findings, and in so doing, we observe that if we consider a linearly ordered value quantale for which the quantale operation distribute over arbitrary meets and the strong de Morgan law holds, then the category of quantale-valued gauge transformation groups is isomorphic to the category of quantale-valued approach convergence transformation groups. This captures, for instance, that if the quantale is extended real half line bestowed with opposite order and extended addition as quantale-operations, then the extended real half line valued gauge transformation groups and the extended real half line valued convergence transformation group coincide. Furthermore, we construct on the group of homeomorphisms, a quantale-valued convergence structure, and study some related properties. We remark that given an arbitrary group, and a quantale-valued convergence space, we can construct a quantale-valued convergence transformation group which enjoys some of the properties similar to the former.
Generalizing the notion of probabilistic convergence ring, where the underlying lattice is a specific quantale, we consider more general quantale to introduce the notions of quantale…
Considering a value quantale, we introduce a category of quantale-valued convergence transformation groups on quantale-valued convergence spaces.
In this article, we focus on discussing some subcategories of the category of probabilistic convergence groups, PConvGrp. In so doing, we introduce a category of probabilistic…