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
1. (a)
  $x\\cdot e=x$ ,
2. (b)
  $e\\cdot x=e$ ,
3. (c)
  $(x\\cdot y)\\cdot z=x\\cdot(y\\cdot z)$ ,
4. (d)
  $x\\cdot x^{-1}=e$ ,
5. (e)
  $x^{-1}\\cdot x=e$ , and
6. (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
1. (a)
  $x\\vee x=x$ ,
2. (b)
  $x\\vee y=y\\vee x$ ,
3. (c)
  $x\\vee(y\\vee z)=(x\\vee y)\\vee z$ ,
4. (d)
  $x\\wedge x=x$ ,
5. (e)
  $x\\wedge y=y\\wedge x$ ,
6. (f)
  $x\\wedge(y\\wedge z)=(x\\wedge y)\\wedge z$ ,
7. (g)
  $x\\vee(y\\wedge x)=x$ ,
8. (h)
  $x\\wedge(y\\vee x)=x$ ,
9. (i)
  $x\\vee(y\\wedge(x\\vee z))=(x\\vee y)\\wedge(x\\vee z)$ ,
10. (j)
  $x\\wedge(y\\vee(x\\wedge z))=(x\\wedge y)\\vee(x\\wedge z)$ ,
11. (k)
  $x\\vee(y\\wedge z)=(x\\vee y)\\wedge(x\\vee z)$ , and
12. (l)
  $x\\wedge(y\\vee z)=(x\\wedge y)\\vee(x\\wedge z)$ .
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.