# 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.

Title identity in a class IdentityInAClass 2013-03-22 16:48:05 2013-03-22 16:48:05 CWoo (3771) CWoo (3771) 7 CWoo (3771) Definition msc 08B99 identity