On absorption’s formula definable semigroups of complete theories

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.