As I understand from this post, mapping from plaintext space to ciphertext space is the fundamental point of all order-preserving encryption. So the only way that we let someone encrypt an arbitrary plaintext is to give him/her this mapping. But, on the other hand, if we give someone this mapping, the encryption breaks because anyone who has access to it can easily decrypt any ciphertext since this mapping is usually reversible.
I am not sure at all that I have understood this correctly. Hence this post. To sum up, my question is: Is there any order-preserving encryption that gives everyone the possibility to encrypt an arbitrary message?