A SEMIGROUP OF THEORIES AND ITS LATTICE OF IDEMPOTENT ELEMENTS
Bekenov M.I. Nurakunov A.M.
March 2021Springer
Algebra and Logic
2021#60Issue 11 - 14 pp.
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∗ 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.
algebraic structure , complete theory , direct product of structures , elementary equivalence , lattice , semigroup , theory
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Gumilyov Eurasian National University, Nur-Sultan, Kazakhstan
Institute of Mathematics, National Academy of Science of the Kyrgyz Republic, Bishkek, Kyrgyzstan
Gumilyov Eurasian National University
Institute of Mathematics
10 лет помогаем публиковать статьи Международный издатель
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