On absorption’s formula definable semigroups of complete theories
Bekenov M. Kassatova A. Nurakunov A.
February 2025Springer Science and Business Media Deutschland GmbH
Archive for Mathematical Logic
2025#64Issue 1107 - 116 pp.
On the set of all first-order complete theories T(σ) of a language σ we define a binary operation {·} by the rule: T·S=Th({A×B∣A⊧TandB⊧S}) for any complete theories T,S∈T(σ). The structure ⟨T(σ);·⟩ forms a commutative semigroup. A subsemigroup S of ⟨T(σ);·⟩ is called an absorption’s formula definable semigroup if there is a complete theory T∈T(σ) such that S=⟨{X∈T(σ)∣X·T=T};·⟩. In this event we say that a theory TabsorbsS. In the article we show that for any absorption’s formula definable semigroup S the class Mod(S)={A∈Mod(σ)∣A⊧T0for someT0∈S} is axiomatizable, and there is an idempotent element T∈S that absorbs S. Moreover, Mod(S) is finitely axiomatizable provided T is finitely axiomatizable. We also prove that Mod(S) is a quasivariety (variety) provided T is an universal (a positive universal) theory. Some examples are provided.
Axiomatizable class , Complete theory , Direct product , Quasivariety , Semigroup , Ultraproduct
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L.N. Gumilev Eurasian National University, 13 Kazhymukan Str., Astana, 010008, Kazakhstan
Karaganda Medical University, 40 Gogol Str., Karaganda, 100024, Kazakhstan
Institute of Mathematics, NAS KR, 265a Chu ave., Bishkek, 720071, Kyrgyzstan
L.N. Gumilev Eurasian National University
Karaganda Medical University
Institute of Mathematics
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