A lay abbot in the narrow sense, is a layman who is entrusted with the conduct and use of a monastery without being a member of its convent or even a monk. In the Frankish Empire of the 9th and 10th centuries and its successor states, members of the nobility were particularly frequently vested with this function. In a secondary meaning, lay abbot also designates the clerical holder of a commendam, who does not have the status of a monk (frequent from the High Middle Ages to the early modern period).
Bibliography
F.J. Felten, Äbte und Laienäbte im Frankenreich, 1980.