Algebraic Functions

Let A be an algebra. We say that the functions f1, . . . , fm : An → A are algebraic on A provided there is a finite system of term-equalities \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\bigwedge t_{k}(\overline{x}, \overline{z}) = s_{k}(\overline{x}, \overline{z})}}$$\end{document} satisfying that for each \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\overline{a} \in A^{n}}}$$\end{document}, the m-tuple \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{(f_{1}(\overline{a}), \ldots , f_{m}(\overline{a}))}}$$\end{document} is the unique solution in Am to the system \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\bigwedge t_{k}(\overline{a}, \overline{z}) = s_{k}(\overline{a}, \overline{z})}}$$\end{document}. In this work we present a collection of general tools for the study of algebraic functions, and apply them to obtain characterizations for algebraic functions on distributive lattices, Stone algebras, finite abelian groups and vector spaces, among other well known algebraic structures.