## Varieties generated by finite algebras

Here we describe how to read in a bunch of algebras and test whether the varieties they generate have certain properties.

We have some files in the UACalc/Algebra directory called `Alg1.ua`

, `Alg2.ua`

,..., `Alg9.ua`

each one containing a finite idempotent algebra, and we want to check whether the varieties generated by these algebras are congruence distributive or congruence permutable.

Freese and Valeriote [1] discovered polynomial time algorithms to test for these properties, and implemented them in the UACalc.

Here's how one could use Jython to call the UACalc methods that perform these tests:

from org.uacalc.alg.Malcev import isCongruenceDistIdempotent from org.uacalc.alg.Malcev import cpIdempotent from org.uacalc.io import AlgebraIO homedir = "/home/williamdemeo/git/UACalc/" outfile1 = open(homedir+"isCD.txt", 'w') outfile2 = open(homedir+"isCP.txt", 'w') for k in range(1,10): algname = "Alg"+str(k) algfile = homedir+"Algebras/"+algname+".ua" A = AlgebraIO.readAlgebraFile(algfile) outfile1.write(algname+" "+str(isCongruenceDistIdempotent(A, None))+"\n") outfile2.write(algname+" "+str((cpIdempotent(A, None)==None))+"\n") outfile1.close() outfile2.close()

The output will be stored in the isCD.txt and isCP.txt files, and will look something like this:

Alg1 True Alg2 True Alg3 False Alg4 True...

with True in the isCD.txt (resp, isCP.txt) file meaning the algebra generates a variety that is congruence distributive (resp, permutable).

## References

[1] Ralph Freese and Matthew Valeriote, On the complexity of some Maltsev conditions, Intern. J. Algebra & Computation, 19(2009), 41-77.