Hypergraphs geometry

One of the problem we are interested to investigate is the representation of hypergraph structures by lower-order structures. This problem can be formulated and formalized as follows.
Problem 1. Given a hypergraph H and its projection P into the graph G, what should be the set of properties of projection P of a hypergraph such that metrics M introduced on the hypergraph H keeps its properties as projected metrics M' on a graph G. By metrics M for a hypergraph H we mean a function defined on a set of nodes of a hypergraph H: M(xi,xj), where xi and xj are elements from the set of nodes X of hypergraph H (X,E) and graph G.  By metrics M' for a graph H we mean a function defined on a set of nodes of a hypergraph H: M(xi,xj), where xi and xj are elements from the set of nodes of hypergraph H and graph G.
This problem can be formalised into a problem of finding the correspondence between all sets of hypergraphs on a set of nodes X and sets of graphs with metrics on the same set of nodes X.

Problem 2 (variation of Problem 1). Given a hypergraph H and its projection P into the lower-arity hypergraph H', what should be the set of properties of projection P of a hypergraph such that metrics M introduced on the hypergraph H keeps its properties as projected metrics M' on a hypergraph H' of a limited arity k. 

We construct a weighted hypergraph given the set of sets of articles using similar construction as for the hypergeometric p-value. We assign a weight to each hyperedge of a a hypergraph, which corresponds to the hypergeometric p-value assigned to the overlapping sets, as described in [3]. 

The definition of the cognitive distance on hypergraphs is hence the generalization of the cognitive distance on graphs (Singh, Tupikina...Santolini et al. 2023 [3]). The properties of cognitive distance can be then studied using the theory of distances on hypergraphs (see above).

Algebras definition is important to remind here, as the first key idea to describe the algebraic structures further. We start with the binary-operation algebras.

Let K be a field, and let A be a vector space over K equipped with an additional binary operation from A × A to A, denoted here by · (that is, if x and y are any two elements of A, then x · y is an element of A that is called the product of x and y). Then A is an algebra over K if the following identities hold for all elements x, y, z in A , and all elements (scalars) a and b in K:

• Right distributivity: (x + y) · z = x · z + y · z

• Left distributivity: z · (x + y) = z · x + z · y

• Compatibility with scalars: (ax) · (by) = (ab) (x · y).

Hypothesis on hypergraphs algebra: 

Each hypergraph is associated to the algebra. Some algebras (e.g. algebras of the polynomials) can be represented as 

the sum of the subalgebras, e.g.  A = A_0 + A_1 +  ... + A_k... A_n, where k is the degree of the polynomial.  

A_d AK \in A_{d+1}, d in Z. This analogy can also be extended to the hyper graphs of specific higher arity in the case of hypergraph Bernstein algebras.
 

Ternary structures play a special role in helping to describe hypergraph structures. Important perspective on ternary structures arrives from the fact that they are not simply generalizations of binary structures. There are, for example, ternary algebras which can be formed not from generalisation of binary ones, but rather be independent examples such as algebra Lie. It is easy to demonstrate that the cardinality of the set of all ternary agebras is larger than the cardinality of the set of all binary ones. Some of this is described in Kerner's work [4], where they considered ternary, Z3-graded generalization of the Heisenberg algebras and found that introducing a non-trivial cubic root of unity, j=e^2πi3, one can define two types of creation operators instead of one, accompanying the usual annihilation operator.