Can anyone help to find some information about these structures? 



Algorithmically finite-dimensional (noncommutative) algebras.
All what I could found is a talk of Bakhadyr Khoussainov


<http://www.math.nsc.ru/conference/malmeet/11/plenary/2011MM_Khoussainov.ppt>


in which he considered "algorithmically finite universal algebras". Maybe this will useful for you.


