Y are distinct fimctions then fT, gT: XT , YT are again distillct ftmctions. Proof· I implies 2. By hypothesis, some T-algebra has at least two elements. 27), for each set X we can construct aT-algebra (Y, 8) and an injective function f: X ----'> Y. 8) = f. ;f_ _ _4 >Y XYJ YYJ X T - - - - - - - - 4 ) YT fT is injective.

T). 3 + ). 14 + that Set(Q, E) may be "identified" with the category of (Q, E)-algebras of form AT. 22 is valid when n = O. Let Q have a single nullary operation and let E be empty. Show that Set(Q, E) may be identified with the category of sets and partial functions. 3. 17. In this section we describe Set(Q, E) as an "algebraic" object without reference to any (Q, E). 10, no intrinsic structure of sets is referred to; we need only to know that sets and functions form a category. 1 Definition. Fix an arbitrary category :f{".

### Algebraic Theories by Ernest G. Manes (auth.)

by Mark

4.2