B we define fT:AT -----+) BT = 0 Proof. (idA)T = id AT 0 (idA)LI = id AT 0 AI] = id AT . gT. D Let us explore this new construction in some previous examples. In the (Q, E) case, fT = id AT 0 f,1 = Cf':l)#.
O = «(xy)~(z)~, 0 = (x*y)*z. Similarly, using (x)(yz), (xyz)~ = x*(y*z). 21. By the time we have finished section 5, it will be clear that T-algebras are always (Q, E)algebras so long as Q is not restricted to finitary operations. The technical convenience of the finitary restriction has been great. 11 and its many successive consequences are much more cumbersome with infinitary formulas and the reader would have perhaps been much confused ifwe had attempted this. Let us devote the rest ofthis section to isolating the "finitary" algebraic theories and proving that they are coextensive with finitary universal algebra.
4. 5. 6. 7. 8. 9. 15 we did not explicitly show that, for a fixed set X, the passage from 6 to ~ is injective. 7. A semilattice is a partially ordered set in wh ich every pair of elements has a supremum. Let T be the algebraic theory of nonempty finite subsets (cf. exercises 7, 10 of section 3). Show that See may be identified with the category of semilattices and functions which preserve binary suprema. 19 is not finitary. Why is "groups" not a variety in "monoids"? Show that a subsemigroup of a group need not be a subgroup even if it is a group.