Sets and Posets with Inversions

We investigate unary operations ∨, ∧ and ♦ on a set X satisfying x = x∨∨ = x∧∧ and x♦ = x∨∧ = x∧∨ for all x ∈ X. Moreover, if in particular X is a meet-semilattice, then we also investigate the operations defined by x = x ∧ x∨, x = x ∧ x∧, x = x ∧ x♦; x• = x∨ ∧ x∧, x♣ = x∨ ∧ x♦, x♠ = x∧ ∧ x♦; and x = x ∧ x∨ ∧ x∧ ∧ x♦ for all x ∈ X. Our prime example for this is the set-lattice P(U × V ) of all relations on one group U to another V equipped with the operations defined such that F ∨(u) = F (−u), F ∧(u) = −F (u) and F ♦(u) = −F (−u) for all F ⊂ U× V and u ∈ U . 1. A few basic facts on relations and functions A subset F of a product set X× Y is called a relation on X to Y . If in particular F ⊂ X2, then we may simply say that F is a relation on X . In particular, ΔX = {(x, x) : x ∈ X} is called the identity relation on X . If F is a relation on X to Y , then for any x ∈ X and A ⊂ X the sets F (x) = { y ∈ Y : (x, y) ∈ F} and F [A] = ⋃a∈A F (a) are called the images of x and A under F , respectively. Instead of y ∈ F (x) sometimes we shall also write xF y. Moreover, the sets DF = { x ∈ X : F (x) = ∅} and RF = F [X ] = F [DF ] will be called the domain and range of F , respectively. If in particular DF = X , then we say that F is a relation of X to Y , or that F is a total relation on X to Y . While, if RF = Y , then we say that F is a relation on X onto Y . 2010 Mathematics Subject Classification: Primary 06A06, 06A11; Secondary 06A12, 20M15.