Re: Properties of a certain algebra


mateus.oliveira@xxxxxxxxx wrote:
> hello, I'd like to know interesting properties of the fallowing
> algebra:
>
> 1) Is finite.
> 2) It's associative: (AB)C = A(BC)
> 3) There is an identity = eA=Ae=A
> 4) every element is idempotent: A=A^2 for any A
> 5) have the following polynomial identity: ABA = AB for any A and B
> 6) No element has inverse, i.e for any A there's no A' such that AA'=e

Take any finite set X with n elements. Let A be the set of all words in
the alphabet X such that no element letter is repeated. Define
multiplication to be concatenation followed by use of 4) and 5) to
remove repetitions.

Example: If X={a,b} you have the example in Prof. Israel's reply.

In general, this A would consist of all permutations of all subsets of
X. Whatever one calls this class, these could be called the free
whatevers. They are clearly not commutative without additional
relations.


Dan

.

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