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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).doi:10.5539/jmr.v4n2p101 fatcat:bshu7migingwpcykqtej6skmui