An Investigation of the Properties of the (p, Q)-superlattice

The (P,Q)-superlattice is a hyperstructure of the form (L, P ∨, Q ∧), where (L,∨,∧) is a lattice and the hyperoperations P ∨, Q ∧ are defined as follows: a P ∨ b . = a ∨ b ∨ P , a Q ∧ b . = a ∧ b ∧ Q. If the sets P,Q ⊆ L satisfy appropriate conditions, then (L, P ∨, Q ∧) is a superlattice. We examine the properties of the sets a P ∨ b and a Q ∧ b in connection to the properties of P,Q. Especially, we consider certain “order-like” relationships between such sets. We also examine various types of P ∨ and Q ∧ distributivity. AMS classification number: 06B99.