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 ST∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {S}_T^{\ast } $$\end{document} by a semigroup ST . 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.