IdentityInAClass 2007-03-05 23:03:57 2007-06-13 13:09:18 Definition equational class identity \usepackage{amssymb,amscd} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions \usepackage{amsthm} % making logically defined graphics \usepackage{xypic} \usepackage{pst-plot} \usepackage{psfrag} % define commands here \newtheorem{prop}{Proposition} \newtheorem{thm}{Theorem} \newtheorem{ex}{Example} \newcommand{\real}{\mathbb{R}} Let $K$ be a class of algebraic systems of the same type. An \emph{identity} on $K$ is an expression of the form $p=q$, where $p$ and $q$ are $n$-ary polynomial symbols of $K$, such that, for every algebra $A\in K$, we have $$p_A(a_1,\ldots, a_n)=q_A(a_1,\ldots,a_n)\qquad\mbox{ for all }a_1,\ldots, a_n\in A,$$ where $p_A$ and $q_A$ denote the induced polynomials of $A$ by the corresponding polynomial symbols. An identity is also known sometimes as an \emph{equation}. \textbf{Examples}. \begin{itemize} \item Let $K$ be a class of algebras of the type $\lbrace e, ^{-1}, \cdot \rbrace$, where $e$ is nullary, $^{-1}$ unary, and $\cdot$ binary. Then \begin{enumerate} \item $x\cdot e=x$, \item $e\cdot x=e$, \item $(x\cdot y)\cdot z=x\cdot (y\cdot z)$, \item $x\cdot x^{-1}=e$, \item $x^{-1}\cdot x=e$, and \item $x\cdot y= y\cdot x$. \end{enumerate} can all be considered identities on $K$. For example, in the fourth equation, the right hand side is the unary polynomial $q(x)=e$. Any algebraic system satisfying the first three identities is a monoid. If a monoid also satisfies identities 4 and 5, then it is a group. A group satisfying the last identity is an abelian group. \item Let $L$ be a class of algebras of the type $\lbrace \vee, \wedge \rbrace$ where $\vee$ and $\wedge$ are both binary. Consider the following possible identities \begin{enumerate} \item $x\vee x=x$, \item $x\vee y=y\vee x$, \item $x\vee (y\vee z)=(x\vee y)\vee z$, \item $x\wedge x=x$, \item $x\wedge y=y\wedge x$, \item $x\wedge (y\wedge z)=(x\wedge y)\wedge z$, \item $x\vee (y\wedge x)=x$, \item $x\wedge (y\vee x)=x$, \item $x\vee (y\wedge (x\vee z))=(x\vee y)\wedge (x\vee z)$, \item $x\wedge (y\vee (x\wedge z))=(x\wedge y)\vee (x\wedge z)$, \item $x\vee (y\wedge z)=(x\vee y)\wedge (x\vee z)$, and \item $x\wedge (y\vee z)=(x\wedge y)\vee (x\wedge z)$. \end{enumerate} If algebras of $K$ satisfy identities 1-8, then $K$ is a class of lattices. If 9 and 10 are satisfied as well, then $K$ is a class of modular lattices. If every identity is satisified by algebras of $K$, then $K$ is a class of distributive lattices. \end{itemize}