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...

A salt metathesis reaction is a chemical process involving the exchange of bonds between two reacting chemical species, which results in the creation of products with similar or identical bonding affiliations. Hyperstructure theory is studied from the theoretical point of view and for its applications. In this paper, we provide some examples of hyperstructures associated with salt m...

In this paper, we have generalized the definition of vector space by considering the group as a canonical $m$-ary hypergroup, the field as a krasner $(m,n)$-hyperfield and considering the multiplication structure of a vector by a scalar as hyperstructure. Also we will be consider a normed $m$-ary hypervector space and introduce the concept of convergence of sequence on $m$-ary hypernormed space...

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 ...

Most of the existing simultaneous approaches for heat exchanger network synthesis assume that only one type of hot/cold utility is available to adjust the final temperatures. In this work, we propose a simultaneous mixed-integer nonlinear programming formulation based on a more generalized stagewise superstructure. We allow utilities to adjust the temperature of process streams in each stage. W...

Hyperstructure theory was born in 1934 when Marty [19] defined hypergroups as a generalization of groups. Let H be a non-empty set and let ℘∗(H) be the set of all non-empty subsets of H. A hyperoperation on H is a map ◦ : H ×H −→ ℘∗(H) and the couple (H, ◦) is called a hypergroupoid. If A and B are non-empty subsets of H, then we denote A◦B = ∪ a∈A, b∈B a◦b, x◦A = {x}◦A and A◦x = A◦{x}. Under c...

in this paper we study two important concepts, i.e. the direct andthe inverse limit of hyperstructures associated with fuzzy sets of type 2, andshow that the direct and the inverse limit of hyperstructures associated withfuzzy sets of type 2 are also hyperstructures associated with fuzzy sets of type 2.

This  paper extends the notion of fuzzy $BCK$-subalgebras to fuzzy hyper $BCK$-subalgebras and defines an extended fuzzy $BCK$-subalgebras. This study  considers  a type of fuzzy hyper  $BCK$-ideals  in this  hyperstructure  and  describes   the  relationship  between hyper  $BCK$-ideals and  fuzzy hyper  $BCK$-ideals.  In fact, it tries to introduce   a  strongly  regular  relation  on hyper $...