In a previous work we have introduced the (P,Q)-superlattice, a hyperstructure of the form (L, P ∨, Q ∧). Here (L,∨,∧) is a lattice and the hyperoperations P ∨, Q ∧ are defined by a P ∨b . = a∨b∨P , a Q ∧b . = a∧b∧Q; when the sets P,Q ⊆ L satisfy appropriate conditions (L, P ∨, Q ∧) is a superlattice. In this work we continue the investigation of (P,Q)superlattice and consider the structure of ...

The largest class of hyperstructures is the one which satisfies the  weak properties. We connect the theory of P-hopes, a large class of  hyperoperations, with the Lie-Santilli admissibility used in  Hardonic Mechanics. This can be achieved by a kind of Ree,  sandwich hyperoperation.

On the the lattice (L,∨,∧) we construct the hyperoperations P ∨, Q ∧ 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. In this paper we give two representation results for (P,Q)-superlattices. The first result is an analog of the [1] representation result. The second result gives necessary and sufficient...

In this paper we study the concept of sets of elements, related to results of an associative binary operation. We discuss this issue in the context of matrices and lattices. First of all, we define hyperoperations similar to those used when constructing hyperstructures from quasiordered semigroups. This then enables us to show that if entries of matrices are elements of lattices, these consider...

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. Especial...

We present a hyperproduct on non square matrices by using a generalization of the well known P-hopes. This theory is connected with the corresponding classical algebra, mainly with the theory of representations by (hyper) matrices. This can be achieved by using the fundamental relations defined on the hyperstructures.

This paper generalizes the idea of strongly associative hyperoperation introduced in [7]  to the class of hyperrings. We introduce and investigate hyperrings of type 1, type 2 and SDIS. Moreover, we study some examples of these hyperrings and give a new kind of hyperrings called  totally hyperrings. Totally hyperrings give us a characterization of Krasner hyperrings. Also, we investigate these ...