identity in a class
Let be a class of algebraic systems of the same type. An identity on is an expression of the form , where and are -ary polynomial symbols of , such that, for every algebra , we have
where and denote the induced polynomials of by the corresponding polynomial symbols. An identity is also known sometimes as an equation.
Examples.
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•
Let be a class of algebras of the type , where is nullary, unary, and binary. Then
-
(a)
,
-
(b)
,
-
(c)
,
-
(d)
,
-
(e)
, and
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(f)
.
can all be considered identities on . For example, in the fourth equation, the right hand side is the unary polynomial . 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.
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(a)
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•
Let be a class of algebras of the type where and are both binary. Consider the following possible identities
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(a)
,
-
(b)
,
-
(c)
,
-
(d)
,
-
(e)
,
-
(f)
,
-
(g)
,
-
(h)
,
-
(i)
,
-
(j)
,
-
(k)
, and
-
(l)
.
If algebras of satisfy identities 1-8, then is a class of lattices. If 9 and 10 are satisfied as well, then is a class of modular lattices. If every identity is satisified by algebras of , then is a class of distributive lattices.
-
(a)
Title | identity in a class |
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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 |