Galois and Pataki Connections for Ordinary Functions and Super Relations

Abstract

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 φR defined by φR (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 6= ∅ } for all B ⊆ X . By using complement and inverse relations, we prove that lbR = cl c Rc and clR (B) = R−1 [B ] . We also consider the dual super relations ubR = lbR−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 = lbR−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) clR−1 is intR – normal, or equivalently clR is intR−1 – normal ; (2) ub c R is lbR ◦ CY – normal, or equivalently lb c R is ubR ◦ CX – normal ; (3) R is a function of X to Y if and only if clR−1 is clR – normal, or equivalently intR is intR−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 6= ∅ .