An example of a positive definite function which is not of positive type on Z2

of positive type is positive definite, and every scalar-valued, positive definite function is of positive type. But a positive definite function is not necessarily of positive type. In fact, T. M. Bisgaard demonstrated that there exists an explicit example of a positive definite function which is not of positive type on (No,+,x*=x) where N0:={0,1,2,...}(see [1, Theorem 1]), and we did on (Z,+,x*=x) (see [3, Theorem 3.7]). For abelian *-semlgroups (N20,+,x*=x) and (Z2,+,x*=x), is there such an example? When (No,+,x*=x), the answer is clear because we have the zero extension of Bisgaard's example (i. e.