Relationship of algebraic theories to powersets over objects in Set and Set × C

This paper deals with a particular question—When do powersets in lattice-valued mathematics form algebraic theories (or monads) in clone form? Our approach in this and related papers is to consider “powersets over objects” in the ground categories Set and Set×C from the standpoint of algebraic theories in clone form (C is a particular subcategory of the dual of the category of semi-quantales). For both fixed-basis powersets over objects of Set and variable-basis powersets over objects of Set×C, necessary and sufficient conditions are found under which the family of all such powersets over a ground object forms an algebraic theory in clone form of standard construction. In such results a distinguished role emerges for unital quantales.