If so, identify the categories. Characterise products in Pfn , assuming excluded middle. Let L be the two-sorted algebraic theory of a commutative ring together with a module. Let L be a single-sorted algebraic theory. Let L be an algebraic theory all of whose operation-symbols happen to be unary.
L qua unary and algebraic theories respectively.
Show that any internal monoid in the category Mon of monoids is commutative. Hence show that the category of complete lattices and Scott- continuous functions is cartesian closed. Let L be a single-sorted algebraic theory. N such that addition is the categorical product. Conversely show that every such category C arises in this way.