This work is devoted to a preliminary investigation of positive Clifford semigroups on the plane. A positive semigroup is a semigroup which has a copy of the nonnegative real numbers embedded as a closed subset in such a way that 0 is a zero and 1 is an identity. A positive Clifford semigroup is a positive semigroup which is the union of groups. In this work it is shown that if S is a positive Clifford semigroup on the plane, then each group in S is commutative. Also, a necessary and sufficient condition is given in order that S be commutative, and an example is given of such a semigroup which is, in fact, not commutative. In addition, both the number and the structure of the components of groups in S is determined. Finally, it is shown that S is the continuous isomorphic image of a semilattice of groups. A topological semigroup is a Hausdorff space together with a continuous associative multiplication. A real semigroup has been defined by J. G. Horne, Jr. [4] to be a topological semigroup containing a subsemigroup R iseomorphic to multiplicative semigroup of real numbers, embedded as a closed subset of E2 in such a way that 1 is an identity and 0 is a zero. Similarly, the author has defined a positive semigroup to be a topological semigroup containing a subsemigroup N iseomorphic to the multiplicative semigroup of nonnegative real numbers, embedded as a closed subset of E2 so that 1 is an identity and 0 is a zero [2]. Relying heavily on the work done by Horne in [4] and [5], this work is devoted to a study of positive semigroups on E2 with the additional requirement that these semigroups be the union of groups. Let us call such semigroups positive Clifford semigroups [3]. We will show that if S is a positive Clifford semigroup on E2, then each group in S is commutative. Also, we will give a necessary and sufficient condition in order that a positive Clifford semigroup on E2 be commutative, and we will give an example of a positive Clifford semigroup on E2 which is, in fact, not commutative. We will show that each group in a positive Clifford semigroup S on E2 has one, two, or four components, that each two dimensional group is P x P, P x P x {1,, or P x P x F, where F is the four group, and that each one dimensional group is P, P x {1, 1}, or P x F. Also, we will characterize S in terms of the sector of identity Presented to the Society, November 8, 1968; received by the editors May 19, 1969 and, in revised form, January 8, 1970. AMS 1969 Subject Classifications. Primary 2205.