The mitzvah of appointing a king has a potential application to women's suffrage (right to vote for you of those in Rio Linda). The Or Sameiach to Rambam Hilchos Melachim 1:5 says:
וזאת מן המצוות המוטלות על הציבור כולן הזכרים, כי להם יאות לעשות ענינים אלה.
ונוהגת מצוה זו בכל מקום ובכל זמן, בזכרים כי להם לעשות המלחמה ונקמת האויב, לא לנשים.
This issue in the early 1920s affected the Yishuv as it was a worldwide issue at that time. Rav Kook paskened (Ma’amarei ha-RaAy’aH pp. 189-194) women should not vote, for reasons like the Or Sameiach and Chinuch, though he surprisingly does not quote them. Rav Uziel paskened that women can vote (Piskei Uziel no. 43-4). These are translated, with a missing page of Rav Kook here (opens in new window as pdf).
(Thus the position offered by Rabbi Marc Angel is somewhat baffling. He writes: Rabbi Kook and Rabbi Uziel also differed in matters pertaining to the role of women in civic life. Their halakhic decisions reflected their different attitudes toward the role of women in a traditional society....the rabbinical leadership of the old Yishuv generally opposed extending to women the right to vote and hold public office. In contrast, the Sephardic rabbinic leadership generally favored granting women those rights. Now, while the authorship of the Chinuch is uncertain, it was definitely written in Spain. So the opinion of Rav Uziel is NOT the traditional Spanish Rabbinical position; the Ashkenazic leaders of the Yishuv in Palestine paskened according to the Chinuch's position, a Spanish sage [though we cannot say his opinion is agreed upon by all Spanish Rishonim]!)