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
$${p}_{A}({a}_{1},\mathrm{\dots},{a}_{n})={q}_{A}({a}_{1},\mathrm{\dots},{a}_{n})\mathit{\hspace{1em}\hspace{1em}}\text{for all}{a}_{1},\mathrm{\dots},{a}_{n}\in A,$$ 
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^{}.

(a)

•
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.

(a)
Title  identity in a class 

Canonical name  IdentityInAClass 
Date of creation  20130322 16:48:05 
Last modified on  20130322 16:48:05 
Owner  CWoo (3771) 
Last modified by  CWoo (3771) 
Numerical id  7 
Author  CWoo (3771) 
Entry type  Definition 
Classification  msc 08B99 
Defines  identity 