Thank you so much for your speedy response and quick fix! I can't thank you 
enough!

On Wednesday, March 29, 2017 at 10:17:10 AM UTC-4, Simon King wrote:
>
> Hi!
>
> Am Mittwoch, 29. März 2017 14:14:28 UTC+2 schrieb Simon King:
>>
>> I will open a trac ticket for it. 
>>
>
> I opened https://trac.sagemath.org/ticket/22707. With the branch from 
> there (that hasn't been tested yet, so, handle with care!), one has
>
> sage: R.<x,y,z> = FreeAlgebra(Integers(2))
> sage: 1 in R
> True
> sage: %timeit (x+y+z)*(x+y+z)
> 10000 loops, best of 3: 99.2 µs per loop
>
> sage: R.<x,y,z> = FreeAlgebra(Integers(2), implementation='letterplace')
> sage: 1 in R
> True
> sage: %timeit (x+y+z)*(x+y+z)
> 10000 loops, best of 3: 31.7 µs per loop
>
> sage: R = 
> DiGraph({1:{1:['x','y','z']}}).path_semigroup().algebra(Integers(2))
> sage: R.inject_variables()
> Defining e_1, x, y, z
> sage: 1 in R
> True
> sage: %timeit (x+y+z)*(x+y+z)
> 100000 loops, best of 3: 4.5 µs per loop
>
>
>
> So, if you care for the speed of arithmetic operations, you shouldn't use 
> the current default implementation of FreeAlgebra. I guess changing the 
> default should be another trac ticket.
>
> Best regards,
> Simon
>

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