Relationship of algebraic theories to powersets over objects in Set
Stephen Ernest Rodabaugh · 2009
Thispaperdealswithaparticularquestion—Whendopowersetsinlattice-valuedmathematicsformalgebraictheories(ormonads) incloneform?Ourapproachinthisandrelatedpapersistoconsider “powersetsoverobjects”inthegroundcategoriesSetandSet×C 9 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 11 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. 13