Characterizations of M-fuzzifying convex spaces via Galois connections

This paper introduces a special Galois connection combined with the wedge-below relation. Furthermore, by using this tool, it is shown that the category of M-fuzzifying betweenness spaces and the category of M-fuzzifying convex spaces are isomorphic and the category of arity-2 M-fuzzifying convex spaces can be embedded in the category of M-fuzzifying interval spaces as a reflective subcategory.

Fuzzy Galois connections on Alexandrov L-topologies

In this paper, we introduce the notion of Galois and dual Galois connections as a topological viewpoint of concept lattices in a complete residuated lattice. Under various relations, we investigate the Galois and dual Galois connections on Alexandrov L-topologies. Moreover, their properties and examples are investigated.

Compatibilities between continuous semilattices

We define compatibilities between continuous semilattices as Scott continuous functions from their pairwise cartesian products to $\{0,1\}$ that are zero preserving in each variable. It is shown that many specific kinds of mathematical objects can be regarded as compatibilities, among them monotonic predicates, Galois connections, completely distributive lattices, isotone mappings with images being chains, semilattice morphisms etc. Compatibility between compatibilities is also introduced, it is shown that conjugation of non-additive real-valued or lattice valued measures is its particular case.

Various operations and right completeness in generalized residuated lattices

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.

Invariance groups of functions and related Galois connections

Invariance groups of sets of Boolean functions can be characterized as Galois closures of a suitable Galois connection. We consider such groups in a much more general context using group actions of an abstract group and arbitrary functions instead of Boolean ones. We characterize the Galois closures for both sides of the corresponding Galois connection and apply the results to known group actions.

Relatively divisible and relatively flat objects in exact categories

We introduce and study relatively divisible and relatively flat objects in exact categories in the sense of Quillen. For every relative cotorsion pair [Formula: see text] in an exact category [Formula: see text], [Formula: see text] coincides with the class of relatively flat objects of [Formula: see text] for some relative projectively generated exact structure, while [Formula: see text] coincides with the class of relatively divisible objects of [Formula: see text] for some relative injectively cogenerated exact structure. We exhibit Galois connections between relative cotorsion pairs in exact categories, relative projectively generated exact structures and relative injectively cogenerated exact structures in additive categories. We establish closure properties and characterizations in terms of the approximation theory.