On the set of all first-order theories T(σ) of similarity type σ, a binary operation {·} is defined by the rule T · S = Th({A × B | A |= T and B |= S}) for any theories T,S ∈ T(σ). The structure 〈T(σ); ⋅〉 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 \( {S}_T^{\ast } \) 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 ≤ defined as T ≤ S iff T · S = S for all T,S ∈ T(σ). Also the set of all idempotent complete theories forms a complete lattice with respect to ≤, which is not necessarily a sublattice of the lattice of idempotent theories.

在相似类型 σ的所有一阶理论T (σ)的集合上，二元运算 {·} 由规则T · S = Th({ A × B | A |= T and B |= S } 定义) 对于任何理论T , S ∈ T (σ)。结构< T ( σ ); ⋅> 形成一个交换半群，称为理论半群。我们证明理论的半群是半群\( {S}_T^{\ast } \)的理想扩展半群S _T. 一个半群理论的所有幂等元素的集合就偏序 ≤ 定义为T ≤ S iff T · S = S对于所有T , S ∈ T (σ)形成一个完整格。此外，所有幂等完备理论的集合形成关于 ≤ 的完备格，它不一定是幂等理论格的子格。