A Semigroup of Theories and Its Lattice of Idempotent Elements

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 ∗ 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.