Page 275 - Textos de Matemática Vol. 46

ON THE LOCALNESS OF THE EMBEDDING OF ALGEBRAS 265
Proof. Let C be an object of C with local A-behaviour. Consider a quasi- identity of Q,
(ui(x) = vi(x), i = 1,...,k) ) (u(x) = v(x)), (2.6)
where x = (x1, . . . , xn), and ui(x), vi(x), u(x) and v(x) are terms on the vari- ables x1,...,xn. Given C 2 C, let c1,...,cn 2 C, and put c = (c1,...,cn). We write
uC (c)
for denoting the element of C obtained from u(x) by replacing every xi by ci, and every operation symbol ✓ 2 ⌃ by the operation ✓C .
Suppose that
uCi (c) = viC(c), i = 1,...,k. We want to prove that then uC (c) = vC (c). Put
X={c1,...,cn}[{uCi (c),i=1,...,k}[{uC(c),vC(c)}.
By hypothesis we have a commutative diagram as in (2.5). Without loss of
generality, we may assume that d is an inclusion map. Then,
h(uDi (c)) = uCi (h(c1), ..., h(cn)) = uCi (c1, ..., cn) = uCi (c),
and, analogously, h(viD(c)) = viC(c). Consequently, h(uDi (c)) = h(viD(c)). Con- sider the subset
Z=X[{uDi (c),i=1,...,k}[{viD(c),i=1,...,k}[{uD(c),vD(c)}
of UD, and let f : D ! A be as in (ii) of Definition 2.6. By hypothesis,
ker(Uf·s)=ker(Uh·s),andsotheequalityh(uDi (c))=h(viD(c))implies f(uDi (c))=f(viD(c)).
And then, since f is a homomorphism,
uAi (f(c1), ..., f(cn)) = viA(f(c1), ..., f(cn)), i = 1,...,k.
Hence, since A satisfies the given quasi-identity (2.6), uA(f(c1), ..., f(cn)) = vA(f(c1), ..., f(cn)).
This is the same as f(uD(c)) = f(vD(c)). But, again by the fact that ker(Uf · s) = ker(U h · s), this implies that
h(uD(c)) = h(vD(c)).
That is, taking into account the commutativity of (2.5), where h is a homo-
morphism and the other two maps are inclusion maps, uC(c) = vC(c).
⇤