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 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.
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M. I. Bekenov is supported by MES RK, project No. AP09259295.
Translated from Algebra i Logika, Vol. 60, No. 1, pp. 3-22, January-February, 2021. https://doi.org/10.33048/alglog.2021.60.101
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Bekenov, M.I., Nurakunov, A.M. A Semigroup of Theories and Its Lattice of Idempotent Elements. Algebra Logic 60, 1–14 (2021). https://doi.org/10.1007/s10469-021-09623-1
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DOI: https://doi.org/10.1007/s10469-021-09623-1