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[parent] identity in a class (Definition)

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

$\displaystyle p_A(a_1,\ldots, a_n)=q_A(a_1,\ldots,a_n)$    for all $\displaystyle a_1,\ldots, a_n\in A,$
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 $ \lbrace e, ^{-1}, \cdot \rbrace$, where $ e$ is nullary, $ ^{-1}$ unary, and $ \cdot$ binary. Then
    1. $ x\cdot e=x$,
    2. $ e\cdot x=e$,
    3. $ (x\cdot y)\cdot z=x\cdot (y\cdot z)$,
    4. $ x\cdot x^{-1}=e$,
    5. $ x^{-1}\cdot x=e$, and
    6. $ 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 $ \lbrace \vee, \wedge \rbrace$ where $ \vee$ and $ \wedge$ are both binary. Consider the following possible identities
    1. $ x\vee x=x$,
    2. $ x\vee y=y\vee x$,
    3. $ x\vee (y\vee z)=(x\vee y)\vee z$,
    4. $ x\wedge x=x$,
    5. $ x\wedge y=y\wedge x$,
    6. $ x\wedge (y\wedge z)=(x\wedge y)\wedge z$,
    7. $ x\vee (y\wedge x)=x$,
    8. $ x\wedge (y\vee x)=x$,
    9. $ x\vee (y\wedge (x\vee z))=(x\vee y)\wedge (x\vee z)$,
    10. $ x\wedge (y\vee (x\wedge z))=(x\wedge y)\vee (x\wedge z)$,
    11. $ x\vee (y\wedge z)=(x\vee y)\wedge (x\vee z)$, and
    12. $ 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.



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Also defines:  identity

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Cross-references: distributive, modular, abelian group, group, monoid, polynomial, right hand side, binary, unary, algebras, equation, induced polynomials, algebra, polynomial symbols, expression, type, algebraic systems, class
There are 58 references to this entry.

This is version 4 of identity in a class, born on 2007-03-05, modified 2007-06-13.
Object id is 9035, canonical name is IdentityInAClass.
Accessed 1963 times total.

Classification:
AMS MSC: 08B99 (General algebraic systems :: Varieties :: Miscellaneous)
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