Galois and Pataki Connections for Ordinary Functions and Super Relations
Santanu Acharjee, Michael Th. Rassias, Árpád Száz · Electronic Journal of Mathematics · 2022
A subset R of a product set X × Y is called a relation on X to Y .A relation U on the power set P (X) to Y is called a super relation on X to Y .The relation R can be identified, to some extent, with the set-valued function ϕ R defined by ϕ R (x) = R (x) = y ∈ Y : (x, y) ∈ R for all x ∈ X, and the union-preserving super relation R defined by R (A) = R [ A ] = a∈A R (a) for all A ⊆ X.By using the relation R , we also define two super relations lbR and clR on Y to X such that lbR (B) = x ∈ X : {x} × B ⊆ R and clR (B) = x ∈ X : R (x) ∩ B = ∅ for all B ⊆ X .By using complement and inverse relations, we prove that lbR = cl c R c and clR (B) = R -1 [ B ] .We also consider the dual super relations ubR = lb R -1 and intR = cl c R • CY .If U is a super relation on X to Y and V is a super relation on Y to X, then having in mind Galois connections and residuated mappings, we say that U is V -normal if, for all A ⊆ X and B ⊆ Y , we have U (A) ⊆ B if and only if A ⊆ V (B) .Thus, if U is V -normal, then by defining Φ = V • U and following Pataki's ideas, we see that U is Φ-regular in the sense that, for all A1 , A2 ⊆ X, we have U (A1 ) ⊆ U (A2 ) if and only if A1 ⊆ Φ (A2) .In this paper, by considering a relator (family of relations) R on X to Y , we investigate normality properties of the more general super relations lbR = R∈R lbR and clR = R∈R clR , and their duals ubR = lb R -1 and intR = cl c R • CY .However, as some applicable results of the paper, we only prove that if R is a relation on X to Y , then the following assertions hold : (1(3) R is a function of X to Y if and only if cl R -1 is clR -normal, or equivalently intR is int R -1 -normal .The closure-interior and the upper-lower-bound Galois connections, established in assertions (1) and (2), are applied in the calculus of relations and the completion of posets, respectively.Some of the implications in assertion (3) require that Y = ∅ .