References

are triples (X,A, τ), where X is a set, A is a localic algebra (an object of LoA) and τ is a subalgebra of AX. Morphisms (X,A, τ) (f,ϕ)−−− → (Y,B, σ) are Set × LoA-morphisms (X,A) (f,ϕ)−−− → (Y,B) such that ϕop ◦ p ◦ f ∈ τ for every p ∈ σ (the so-called continuity). Our definition subsumes the traditional lattice-valued 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. Follow-ing 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-called satisfaction relation) such that B |=(x,−)−−−−− → A is a homomorphism (a mor