On Regular Co-Medial Algebras

In this paper we define the concepts of co-medial algebra and regular algebra and we will show that for a regular co-medial algebra $(A,f,g)$, there exist a commutative semigroup $(A,+)$, such that the operations $f,g$ have the linear representation on $(A,+)$. As a consequence of above result, we have the linear representation of an $n-$ary medial groupoid with an regular element, which was obtained by (Evans, T., 1963, p. 331-349).