Proof. The case A = ⊥ is trivial. The remaining case is when A is an aggregate.

Consider any possible conjunctive term HI (where I ⊆ {1, . . . , n}) of Aˆ:

$$(\bigwedge_{i\in I}F_{i})\to(\bigvee_{i\in\overline{{{I}}}}F_{i}).$$
$\boxed{\begin{array}{c}\blacksquare\end{array}}$
For each set X of atoms there is exactly one set I such that X 6|= HI : the set IX that consists of the i's such that X |= Fi. Consequently, for every set X of atoms,

$$\begin{array}{r l}{X\models{\hat{A}}}&{{}{\mathrm{iff}}\quad H_{I_{X}}{\mathrm{~is~not~a~conjunctive~term~of~}}{\hat{A}}}\\ {{}}&{{}{\mathrm{iff}}\quad o p(\{w_{i}\,:\,i\in I_{X}\})\prec N}\\ {{}}&{{}{\mathrm{iff}}\quad o p(\{w_{i}\,:\,X\models F_{i}\})\prec N}\\ {{}}&{{}{\mathrm{iff}}\quad X\models A.}\end{array}$$

Lemma 11. For any aggregate A and any set X of atoms, AˆX *is classically equivalent to* AˆX .

Proof. **Case 1**: X 6|= A. Then AˆX = ⊥ˆ = ⊥. On the other hand, by Lemma 10, X 6|= Aˆ so that AˆX = ⊥ also. **Case 2**: X |= A. Then A is an aggregate, and, by the definition of a reduct, AˆX is

I⊆{1,...,n} : op({wi : i∈I})6≺N  ^ i∈I F X i→ _ i∈I F X i. (39) ^
On the other hand, AˆX is classically equivalent, by Lemma 2, to

$\hat{A}X_{\mu}$. 
$$I{\subseteq}\{1,\ldots,n\}:o p(\{w_{i}:i{\in}I\}){\not\in}N\quad i{\in}I\quad\quad\quad i{\in}T$$
Notice that, since X |= Aˆ by Lemma 10, all implications in the formula above are satisfied by X. Consequently, AˆX is classically equivalent to

$$\bigwedge_{I\subseteq\{1,\ldots,n\}\,:\,o p(\{w_{i}:i\in I\})\neq N}\bigl(\bigl(\bigwedge_{i\in I}F_{i}\bigr)^{X}\to\bigl(\bigvee_{i\in\overline{{{I}}}}F_{i}\bigr)^{X}\bigr),$$
$$\mathrm{{\sin,\;to\;(39).}}$$

and then, by Lemma 2 again, to (39).

Proposition 12. Let A be an aggregate of the form (6) and let G be the corresponding formula (9). Then
(a) G is classically equivalent to A*, and*