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 )a ndF ♦ (u )= −F (−u) for all F ⊂ U × V and u ∈ U.