A semigroup of theories and its lattice of idempotent elements

On the set of all first-order theories $T(\sigma)$ of similarity type $\sigma$, a binary operation $\{\cdot\}$ is defined by the rule $T\cdot S= {\rm Th}(\{A\times B\mid A\models T$ and $B\models S\})$ for any theories $T, S\in T(\sigma)$. The structure $\langle T(\sigma);\cdot\rangle$ forms a commutative semigroup, which is called a semigroup of theories.
We prove that a semigroup of theories is an ideal extension of a semigroup $S^*_T$ by a semigroup $S_T$. The set of all idempotent elements of a semigroup of theories forms a complete lattice with respect to the partial order $\leq$ defined as $T\leq S$ iff $T\cdot S=S$ for all $T, S\in T(\sigma)$. Also the set of all idempotent complete theories forms a complete lattice with respect to $\leq$, which is not necessarily a sublattice of the lattice of idempotent theories.