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∈K, we have
pA(a1,…,an)=qA(a1,…,an) for all a1,…,an∈A, |
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
-
(a)
x⋅e=x,
-
(b)
e⋅x=e,
-
(c)
(x⋅y)⋅z=x⋅(y⋅z),
-
(d)
x⋅x-1=e,
-
(e)
x-1⋅x=e, and
-
(f)
x⋅y=y⋅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 {∨,∧} where ∨ and ∧ are both binary. Consider the following possible identities
-
(a)
x∨x=x,
-
(b)
x∨y=y∨x,
-
(c)
x∨(y∨z)=(x∨y)∨z,
-
(d)
x∧x=x,
-
(e)
x∧y=y∧x,
-
(f)
x∧(y∧z)=(x∧y)∧z,
-
(g)
x∨(y∧x)=x,
-
(h)
x∧(y∨x)=x,
-
(i)
x∨(y∧(x∨z))=(x∨y)∧(x∨z),
-
(j)
x∧(y∨(x∧z))=(x∧y)∨(x∧z),
-
(k)
x∨(y∧z)=(x∨y)∧(x∨z), and
-
(l)
x∧(y∨z)=(x∧y)∨(x∧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.
-
(a)
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 |