What is the female equivalent of Duke?
A female Duke can be a Duke or Duchess. The wife of a Duke is a
Duke is unique amongst the British titles in that if a woman
holds the title in her own right the title may be the same as the
male (unlike, say, Baron, where a woman given the same title is
always a Baroness).
So while Kate Middleton is the Duchess of Cambridge (because she
is married to the Duke - William), the Queen is the Duke of
Lancaster (because she holds the title, not Philip). The Queen is
also a Duchess as a result of being married to a Duke.