Quantale-Valued Tolerance Convergence Function Spaces: Topological and Cartesian-Closed Categorical Perspectives
- 1Department of Mathematics, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia
Abstract
Let Q be an integral, commutative quantale with a completely distributive underlying lattice. It is known that there is a one-to-one correspondence between the category of quantale-valued tolerance spaces and tolerance-preserving mappings, denoted by Q-TOL, and the category of quantale-valued tolerance convergence spaces and continuous mappings, denoted by Q-TOLCONV. Our motivations in this article include introducing Q-tolerance structures on quantale-valued tolerance function spaces as well as on quantale-valued tolerance convergence function spaces. We prove that the category Q-TOLCONV and the category Q-TOLCONVGRP, consisting of quantale-valued tolerance convergence groups, are topological categories, and we provide examples of natural transformations in the category Q-TOL. Moreover, we show that both categories are cartesian closed; in particular, for the category Q-TOLCONV, we present a categorical proof of cartesian closedness via an adjoint functor. Furthermore, we study quantale-valued tolerance convergence function space structures on groups of homeomorphisms, where the homeomorphism group serves as a canonical framework for analyzing the compatibility of quantale-valued convergence with topological symmetries. This analysis clarifies how quantale-valued convergence interacts with algebraic symmetry and provides the structural foundation for our final result that quantale-valued tolerance convergence groups are group objects in Q-TOLCONV.
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BibTeX
@article{ahsanullah2026quantalevalued,
title = {{Quantale-Valued Tolerance Convergence Function Spaces: Topological and Cartesian-Closed Categorical Perspectives}},
author = {T. M. G. Ahsanullah and Jawaher Al-Mufarrij},
journal = {Axioms},
year = {2026},
month = nov,
doi = {10.3390/axioms15110823},
url = {https://www.mdpi.com/2075-1680/15/11/823},
note = {CC BY 4.0},
}
RIS
TY - JOUR
TI - Quantale-Valued Tolerance Convergence Function Spaces: Topological and Cartesian-Closed Categorical Perspectives
AU - T. M. G. Ahsanullah
AU - Jawaher Al-Mufarrij
PY - 2026
DA - 2026/11/05
JO - Axioms
DO - 10.3390/axioms15110823
UR - https://www.mdpi.com/2075-1680/15/11/823
AB - Let Q be an integral, commutative quantale with a completely distributive underlying lattice. It is known that there is a one-to-one correspondence between the category of quantale-valued tolerance spaces and tolerance-preserving mappings, denoted by Q-TOL, and the category of quantale-valued tolerance convergence spaces and continuous mappings, denoted by Q-TOLCONV. Our motivations in this article include introducing Q-tolerance structures on quantale-valued tolerance function spaces as well as on quantale-valued tolerance convergence function spaces. We prove that the category Q-TOLCONV and the category Q-TOLCONVGRP, consisting of quantale-valued tolerance convergence groups, are topological categories, and we provide examples of natural transformations in the category Q-TOL. Moreover, we show that both categories are cartesian closed; in particular, for the category Q-TOLCONV, we present a categorical proof of cartesian closedness via an adjoint functor. Furthermore, we study quantale-valued tolerance convergence function space structures on groups of homeomorphisms, where the homeomorphism group serves as a canonical framework for analyzing the compatibility of quantale-valued convergence with topological symmetries. This analysis clarifies how quantale-valued convergence interacts with algebraic symmetry and provides the structural foundation for our final result that quantale-valued tolerance convergence groups are group objects in Q-TOLCONV.
SN - 2075-1680
ER -