Variable-basis topological systems versus variable-basis topological spaces

LoA) and τ is a subalgebra of A . Morphisms (X,A, τ) (f,φ) −−−→ (Y,B, σ) are Set × LoA-morphisms (X,A) (f,φ) −−−→ (Y,B) such that φ ◦ p ◦ f ∈ τ for every p ∈ σ (the so-called continuity). Our definition subsumes the traditional latticevalued approach of [2]. The motivation for the new concept was provided by the problem of doing fuzzy mathematics without order. In [1] the authors consider a relation between topological systems in the sense of [4] and variable-basis topological spaces over the category of locales. Following the example one can introduce the category LoA-TopSys of variable-basis topological systems over localic algebras. Its objects are tuples (X,A,B, |=), where X is a set, A and B are localic algebras and X×B |= −→ A is a map (the so-