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 SetxC 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 SetxC, 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.

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