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 $\lbrace e, ^{-1}, \cdot \rbrace$ , where $e$ is nullary, $^{-1}$ unary, and $\cdot$ binary. Then
  1. $x\cdot e=x$ ,
  2. $e\cdot x=e$ ,
  3. $(x\cdot y)\cdot z=x\cdot (y\cdot z)$ ,
  4. $x\cdot x^{-1}=e$ ,
  5. $x^{-1}\cdot x=e$ , and
  6. $x\cdot y= y\cdot x$ .
  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 $\lbrace \vee, \wedge \rbrace$ where $\vee$ and $\wedge$ are both binary. Consider the following possible identities
  1. $x\vee x=x$ ,
  2. $x\vee y=y\vee x$ ,
  3. $x\vee (y\vee z)=(x\vee y)\vee z$ ,
  4. $x\wedge x=x$ ,
  5. $x\wedge y=y\wedge x$ ,
  6. $x\wedge (y\wedge z)=(x\wedge y)\wedge z$ ,
  7. $x\vee (y\wedge x)=x$ ,
  8. $x\wedge (y\vee x)=x$ ,
  9. $x\vee (y\wedge (x\vee z))=(x\vee y)\wedge (x\vee z)$ ,
  10. $x\wedge (y\vee (x\wedge z))=(x\wedge y)\vee (x\wedge z)$ ,
  11. $x\vee (y\wedge z)=(x\vee y)\wedge (x\vee z)$ , and
  12. $x\wedge (y\vee z)=(x\wedge y)\vee (x\wedge z)$ .
  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.