On the set of all first-order complete theories T(sigma)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T(\sigma )$$\end{document} of a language sigma\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma $$\end{document} we define a binary operation {<middle dot>}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{\cdot \}$$\end{document} by the rule: T<middle dot>S=Th({AxB divided by A & models;TandB & models;S})\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T\cdot S= {{\,\textrm{Th}\,}}(\{A\times B\mid A\models T \,\,\text {and}\,\, B\models S\})$$\end{document} for any complete theories T,S is an element of T(sigma)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T, S\in T(\sigma )$$\end{document}. The structure < T(sigma);<middle dot>>\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\langle T(\sigma );\cdot \rangle $$\end{document} forms a commutative semigroup. A subsemigroup S of < T(sigma);<middle dot>>\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\langle T(\sigma );\cdot \rangle $$\end{document} is called an absorption's formula definable semigroup if there is a complete theory T is an element of T(sigma)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T\in T(\sigma )$$\end{document} such that S=<{X is an element of T(sigma)divided by X<middle dot>T=T};<middle dot>>\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S=\langle \{X\in T(\sigma )\mid X\cdot T=T\};\cdot \rangle $$\end{document}. 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 is an element of Mod(sigma)divided by A & models;T0for someT0 is an element of S}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\,\textrm{Mod}\,}}(S)=\{A\in {{\,\textrm{Mod}\,}}(\sigma )\mid A\models T_0\,\,\text {for some}\,\, T_0\in S\}$$\end{document} is axiomatizable, and there is an idempotent element T is an element of S\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T\in S$$\end{document} that absorbs S. Moreover, Mod(S)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\,\textrm{Mod}\,}}(S)$$\end{document} is finitely axiomatizable provided T is finitely axiomatizable. We also prove that Mod(S)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\,\textrm{Mod}\,}}(S)$$\end{document} is a quasivariety (variety) provided T is an universal (a positive universal) theory. Some examples are provided.
