What would you like to know about this please?
Yes, they were I'm afraid.
The criteria for making an arrest is very low. They only need the barest suspicion to justify an arrest. The fact that they have received an allegation is sufficient.
In fact, the case law is increasingly developing in the direction that any allegation must be considered and there must be a substantial reason for not making an arrest.
The fact that the allegation was without merit does not retrospectively undo their power of arrest I'm afraid.
Can I clarify anything for you?
Yes, they do that sometimes.
It isn't particularly ethical but that doesn't render the arrest unlawful.