Abstract

On the set of all first-order complete theories $$T(\sigma )$$ of a language $$\sigma $$ we define a binary operation $$\{\cdot \}$$ by the rule: $$T\cdot S= {{\,\textrm{Th}\,}}(\{A\times B\mid A\models T \,\,\text {and}\,\, B\models S\})$$ for any complete theories $$T, S\in T(\sigma )$$. The structure $$\langle T(\sigma );\cdot \rangle $$ forms a commutative semigroup. A subsemigroup S of $$\langle T(\sigma );\cdot \rangle $$ is called an absorption’s formula definable semigroup if there is a complete theory $$T\in T(\sigma )$$ such that $$S=\langle \{X\in T(\sigma )\mid X\cdot T=T\};\cdot \rangle $$. In this event we say that a theory T absorbs S. In the article we show that for any absorption’s formula definable semigroup S the class $${{\,\textrm{Mod}\,}}(S)=\{A\in {{\,\textrm{Mod}\,}}(\sigma )\mid A\models T_0\,\,\text {for some}\,\, T_0\in S\}$$ is axiomatizable, and there is an idempotent element $$T\in S$$ that absorbs S. Moreover, $${{\,\textrm{Mod}\,}}(S)$$ is finitely axiomatizable provided T is finitely axiomatizable. We also prove that $${{\,\textrm{Mod}\,}}(S)$$ is a quasivariety (variety) provided T is an universal (a positive universal) theory. Some examples are provided.