Transformation Digroups

We introduce the notion of a transformation digroup and prove that every digroup is isomorphic to a transformation digroup. The purpose of this paper is to show how to choose a class of non-bijective transformations on a Cartesian product of two sets to define a transformation digroup on the Cartesian product. The main result of this paper is that every digroup is isomorphic to a transformation digroup. The notion of a digroup we shall use in this paper was introduced in Chapter 6 of [6]. Its special case, which is the notion of a digroup with an identity, was introduced independently by people who work in different areas of mathematics ([1], [3] and [5]). After reviewing some basic definitions about digroups in Section 1, we introduce in Section 2 the notion of a symmetric digroup on the Cartesian product ∆ × Γ, where ∆ and Γ are two sets. If |∆| = 1, then a symmetric digroup on ∆×Γ becomes the symmetric group on the set Γ , where |∆| denotes the cardinality of ∆. In Section 3 we prove that every digroup is isomorphic a subdigroup of a symmetric digroup, which is a better counterpart of Cayley’s Theorem in the context of digroups. 1 The Notion of a Digroup The following definition of a digroup is a version of Definition 6.1 of [6]. Definition 1.1 A nonempty set G is called a digroup if there are two binary operations ⇀ · and ↼ · on G such that the following three properties are satisfied. (i) (The Diassociative Law) The two operations ⇀ · and ↼ · are diassociative; that is, x ⇀ · (y ⇀ · z) = (x ⇀ · y) ⇀ · z = x ⇀ · (y ↼ · z), (1)