In the world of mathematical generalizations, generalized groups hold a certain fascination, introducing ideas that diverge from the well-established structure of groups. One characteristic that stands out is the existence of an identity for each element. Building on these intriguing concepts, we delve into the understanding of generalized modules, a structure birthed from the union of generalized group ideas and the operation of module action. In this article, we also take a look at generalized module groupoids, extending the purview of generalized groupoids.

1. Introduction

The generalized groups proposition by Molaei opens doors to an intriguing aspect of groups; while a group has a unique identity element, a generalized group offers an identity for each element instead. On the other hand, groupoids offer yet another perspective to understand groups. A groupoid extends the concept of groups by including a small category where each morphism is invertible. First defined by Brandt in 1926, groupoids have inspired mathematicians towards various objectives, including the idea of 'structured groupoid'.

In this context, this article introduces the idea of module action operating on a generalized group which results in a structure we refer to as a generalized module. We discuss examples of generalized modules and explore their notable properties. Moreover, drawing parallels from generalized groupoids, we discuss the generalized module groupoids over a ring and further talk about the relationship between the category of generalized modules and that of generalized module groupoids.

2. Preliminaries

Before we delve deeper, let's take a moment to quickly recap the basics that lay the groundwork for our discussion further. We touch upon the definitions of categories, groupoids, generalized groups, and bring to light some interesting properties of these structures.

A Category C comprises of certain important components: a class called the class of vertices or objects νC; a class of disjoint sets C (a, b) one for each pair (a, b) ∈νC ×νC, where an element f ∈ C is called a morphism from a to b, written f: a→b; a map ◦:C (a, b) × C (b, c)→ C (a, c) given by (f, g)7→f◦g representing the composition of morphisms in C.

A groupoid G= (νG, G) is referred to as a small category if all the morphisms f, g ∈ G have cod f=dom g, which means fg ∈ Gand that every morphism is an isomorphism. If for all a ∈νG, G(a, a)6=φ, then the groupoid G is said to be connected.

Moving towards generalized groups, we see that it is a non-empty set G equipped with a binary operation, called multiplication, which is governed by a certain set of rules: (ab)c=a(bc) for all a, b, c ∈G; for each a ∈G there exists a unique e(a) ∈G with ae(a)=e(a)a=a; and for each a ∈G, an inverse a⁻¹ ∈G exists such that aa⁻¹=a⁻¹a=e(a).

2.1. Generalized Groups and Generalized group groupoids

A generalized group is said to be a normal generalized group if the identity of the product of any two elements (a,b) is the product of their individual identities. A non-empty subset H of a generalized group G is a generalized subgroup of G if and only if for all elements a, b ∈H, ab^-1 ∈H.

2.2. Generalized Groups from a connected groupoid

We explore the possibility of constructing a generalized group from a connected groupoid. We initiate this process with a connected groupoid equipped with partial composition (◦). Following this choice, we defined an addition on this groupoid using a certain protocol. This addition also displays associativity. With the defined addition operation, every generalized group can be considered a normal generalized group. Therefore, the connection between a generalized group and a connected groupoid can be drawn effectively.