Revisiting semigroups theory and its relations to geometry

Algebraic structures, such as semigroups, groups, algebras previously were considered as a branch of more abstract mathematics. With appearance of algebraic geometry, algebraic complex structures used throughout other areas of mathematics this started to change.
With a recent increasing interest in hypergraph structures (see e.g. papers on this here), these mathematical objects have developed various ways of describing higher-order interactions for complex systems.
Here we are discuss ways to characterize on one hand the algebraic language which could unfold some of the properties of hypergraph structures, and on another hand, to deepen our understanding of the physical characteristics of hypergraphs, in particular, the irreducibility properties of higher order representations to the lower order ones.

Defining algebraic language of semigraphs and relation to hypergraphs

The notion of semigroup was introduced by Suchkevich (mathematician who finished his dissertation in Kharkiv in 1918, and submitted it later in Voronezh State University in 1922). The definition of the semigroup is quite straightfoward if you know some basics of the higher algebra and familiar with the notion of the group theory.
Definition. A semigroup is a fundamental algebraic structure in mathematics. It consists of a non-empty set together with an associative binary operation. Formally, a semigroup is defined as follows:

Let S be a non-empty set, and let * be a binary operation on S. The binary operation * is said to form a semigroup if it satisfies the associative property: for all elements a, b, and c in S: (a * b) * c = a * (b * c)

For example, the set of natural numbers forms a group. But what does geometry have to do with it? For understanding this one needs to further read one of the early semigroup theoretical result stating that every semigroup may be embedded in the semigroup of transformations of some set (analogue of Cayley’s Theorem for groups!!). Let us first analyze and look at it more.
If for instance we have a set denoted by b and semigroup of transformations of such set B. For every semigroup S we can find such a semigroup of transformations B, corresponding (e.g. geometric transformations), such that semigroup S is embedded inside B.

Definition. A groupoid consists of a non-empty set G and a binary operation (*) defined on G. Unlike a group, the operation in a groupoid need not be defined for all pairs of elements in G. Instead, for any elements a and b in G, if the operation (a * b) is defined, we say that there is a product between a and b. Additionally, for each element a in G, there exists an identity element e in G such that the product (a * e) and (e * a) are both defined and equal to a.

One of the ideas behind connecting geometric and algebraic structures is that in some cases to give proofs is more suitable for purely geometric structures, while for some cases, it is easier for algebraic structure, which are essentially the same objects, but written in algebraic language.

Semigroups and groupoid languages prepares us to understand the notion of semiheaps, which can support the description of hypergraph operations (Zapata-Carratala et al. 2022).

Relation of hypergraphs to infinite Abelian groups

Related theories and results on this topic include theory of infinite Abelian groups. In early works of Pruefer who studies Abelian groups he found that any Abelian group forms a Schar, which is an algebraic structure with ternary operations such as
AB^-1A = A
AB^-1C = CB^-1A
(AB^-1C)D-1E =A(B^-1CD-1)E
For more details and further references we recommend to go to Wagner theory in Hollings book.

Question: given an infinite Abelian group how to find Schar structure with ternary operations (semiheaps) inside it?

Relation to Erlangen Program

The important relation is connection to the differential geometric literature on Erlangen Program, which had the main idea that for description of some geometric structures one needs to develop more general algebraic structures, Riemanian geometries being one example of such structures.

The basic idea of Erlangen Program is that the geometric properties of a space are preserved under certain transformations, and these transformations form a group.

By studying the group of transformations associated with a given geometric structure, one can reveal the underlying algebraic structure that characterizes the geometry.

Euclidean Geometry: In Euclidean geometry, the group of transformations preserving lengths, angles, and parallelism is the Euclidean group (E(n)). This group consists of translations, rotations, and reflections. The corresponding algebraic structure is the real vector space R^n, equipped with vector addition and scalar multiplication.

Affine Geometry: Affine transformations preserve parallel lines and ratios of lengths along parallel lines. The group of transformations is the affine group (Aff(n)). The algebraic structure associated with affine geometry is the affine space. Reltion to hypergraphs is described in another example (article).

Question: There are many open questions related to physical properties and invariants of hypergraphs.
The problem of finding the physical invariants for hypergraph structures is not yet clearly defined. General question (highlighted in recent article by researchers here ) is to link the n-arity properties of hypergraphs and partition function properties. The importance and explanations of this are brilliantly explained in their article “Beyond Binary: Hypermatrix Algebra and Irreducible Arity in Higher-Order Systems, 2023”.