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identity in a class
Let $K$ be a class of algebraic systems of the same type. An 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 equation.
Examples.

Let $K$ be a class of algebras of the type $\{e,^{{1}},\cdot\}$, where $e$ is nullary, ${}^{{1}}$ unary, and $\cdot$ binary. Then
(a) $x\cdot e=x$,
(b) $e\cdot x=e$,
(c) $(x\cdot y)\cdot z=x\cdot(y\cdot z)$,
(d) $x\cdot x^{{1}}=e$,
(e) $x^{{1}}\cdot x=e$, and
(f) $x\cdot y=y\cdot x$.
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.

Let $L$ be a class of algebras of the type $\{\vee,\wedge\}$ where $\vee$ and $\wedge$ are both binary. Consider the following possible identities
(a) $x\vee x=x$,
(b) $x\vee y=y\vee x$,
(c) $x\vee(y\vee z)=(x\vee y)\vee z$,
(d) $x\wedge x=x$,
(e) $x\wedge y=y\wedge x$,
(f) $x\wedge(y\wedge z)=(x\wedge y)\wedge z$,
(g) $x\vee(y\wedge x)=x$,
(h) $x\wedge(y\vee x)=x$,
(i) $x\vee(y\wedge(x\vee z))=(x\vee y)\wedge(x\vee z)$,
(j) $x\wedge(y\vee(x\wedge z))=(x\wedge y)\vee(x\wedge z)$,
(k) $x\vee(y\wedge z)=(x\vee y)\wedge(x\vee z)$, and
(l) $x\wedge(y\vee z)=(x\wedge y)\vee(x\wedge z)$.
If algebras of $K$ satisfy identities 18, 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.
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