identity in a class


Let K be a class of algebraic systems of the same type. An identityPlanetmathPlanetmathPlanetmath 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 algebraMathworldPlanetmathPlanetmathPlanetmath AK, we have

pA(a1,,an)=qA(a1,,an)   for all a1,,anA,

where pA and qA 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,}, where e is nullary, -1 unary, and binary. Then

    1. (a)

      xe=x,

    2. (b)

      ex=e,

    3. (c)

      (xy)z=x(yz),

    4. (d)

      xx-1=e,

    5. (e)

      x-1x=e, and

    6. (f)

      xy=yx.

    can all be considered identities on K. For example, in the fourth equation, the right hand side is the unary polynomialMathworldPlanetmath 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 groupMathworldPlanetmath.

  • Let L be a class of algebras of the type {,} where and are both binary. Consider the following possible identities

    1. (a)

      xx=x,

    2. (b)

      xy=yx,

    3. (c)

      x(yz)=(xy)z,

    4. (d)

      xx=x,

    5. (e)

      xy=yx,

    6. (f)

      x(yz)=(xy)z,

    7. (g)

      x(yx)=x,

    8. (h)

      x(yx)=x,

    9. (i)

      x(y(xz))=(xy)(xz),

    10. (j)

      x(y(xz))=(xy)(xz),

    11. (k)

      x(yz)=(xy)(xz), and

    12. (l)

      x(yz)=(xy)(xz).

    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