Hi ‐ ‐ Association, free association and library systems are typically not‐for‐profit corporations that are guided by the Not‐for‐ Profit Corporation Law. However, in the case of a library board of trustees, as you know, section 260‐a of the Education Law brings them within the coverage of the Open Meetings Law. Section 260‐a states in part that meetings of board of trustees "shall be held in conformity with and in pursuance to the provisions" of the Open Meetings Law. Under that law and section 41 of the General Construction Law (which deals with the manner in which various terms are to be construed), a quorum is a majority of the total membership of a board, notwithstanding absences or vacancies, and a board may exercise its authority, powers and duty only by means of an affirmative vote of a majority of its total membership.
I point out, too, that members of boards may be counted toward a quorum and vote by phone when the Not‐for‐Profit Corporation Law applies. However, under the Open Meetings Law, meetings may be validly held only when members are physically present or when they participate via videoconferencing that enables those in two or more locations to observe one another. If a meeting is conducted by means of videoconference, notice of a meeting must indicate that to be so, the locations where the members will participate, and that the public may attend at any of those locations. When those requirements are met, the members are counted for purposes of a quorum, and all may vote.
I hope that I have been of assistance.