nß is a bijection if n is finite. 12. 2 stressed (-)# as more basic than 0, and an alternate definition is easily given. 12. 8 and 0(# = id AT ° 0(. 12. 4. Tbe Algebras of a Tbeory The example of(D, E)-algebras raises the question if an arbitrary algebraic theory T in a category :% has algebras. 17 teaches us that if T is coextensive with its algebras, the way to describe them is as a category "of :%-objects with structure"; specifically, for each object A of :% we should provide a set {e} of T -algebra structures ~ on A and, more important, we should define when a :%-morphismf:A ~B is a T-homomorphism from the T-algebra (A, ~) to the T-algebra (B, 8).

Is the structure map of XT. 7. f = J = f. e. For any functionJ, ~,fT, and e are always Q-homomorphisms. Thus ifJis also an Q-homomorphism, the diagram commutes. 8 Definition. Let Yt' be an arbitrary category and let T be an algebraic theory in Yt'. AT-algebra is a pair (X, ~) where X is an object oJ Yt' and 35 4. 10 below. g:(X, ~) ) (X", C) 1S a T-homomorphism so long as f:(X, ~) ) (X', (') and g:(X', ~') ) (X", C) are. This gives us a category %T of T-algebras and T-homomorphisms and a "forgetful %-object" functor UT:%T ) %.

For the following three exercises (implicit in [Birkhoff'35, page 141J) fix an algebraic theory T in Set. A variety in SetT is a collection of Talgebras closed under the formation of products, subalgebras, and quotients. l'. Given (X, ~; A) where (X, ~) is aT-algebra and A is a subset of X such that