Homomorphisms of nearrings of continuous functions from topological spaces into the asymmetric nearring

There is a unique (up to isomorphism) topological nearring N, whose additive group is the two-dimensional Euclidean group, which has an identity but is not zero symmetric. For any topological space X, we denote by N(X) the nearring of all continuous functions from X to N where the operations on N(X) are the pointwise operations. We determine all the homomorphisms from the nearring N(X) into N(Y) when X is realcompact and Y is completely regular and Hausdorff. This result is then used to show that if both X and Y are either compact and Hausdorff or realcompact generated spaces then the endomorphism semigroups of N(X) and N(Y) are isomorphic if and only if the spaces X and Y are homeomorphic.