Affiliations: Department of Mathematics, Gangneung-Wonju National University, Gangneung, Gangwondo, Korea
Correspondence:
[*]
Corresponding author. Yong Chan Kim, Department of Mathematics, Gangneung-Wonju National University, Gangneung, Gangwondo 25457, Korea. E-mail: [email protected].
Abstract: In this paper, based on generalized residuated lattices as an extension of Zhang’s complete residuated lattices, there are two types of structures: bi-partially orders, right (left) joins, right (left) complete lattices and right (left) Alexandrov topologies. We investigate their properties and the relationship between them. Moreover, monotone maps, right (left)-embedding maps and right-join (left-join) preserving maps are investigated with various operations as extensions of Zadeh powerset operations between these structures. As the foundation of fuzzy rough sets and fuzzy contexts, there exist adjunctions and Galois connections between maps from right(left) Alexandrov topologies to right(left) Alexandrov topologies. We give their examples.
Keywords: Generalized residuated lattices, right (left) joins, right (left) meets, right (left) complete lattices, right(left) Alexandrov topologies, adjunctions, Galois connections
