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
Canonical name	IdentityInAClass
Date of creation	2013-03-22 16:48:05
Last modified on	2013-03-22 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