Rajoittumaesijärjestys osittaisille latinalaisneliötensoreille

The rank of a tensor is a fundamental measure of its computational complexity and expressive power, central to the study of bilinear algorithms, such as matrix multiplication, where computational problems are modelled as tensors. However, for comparing tensors, rank is often too coarse. A more fine-grained approach is given by the notion of restriction, which define preorders on tensors. For example, larger matrix multiplication tensors restrict to the Kronecker product of three smaller tensors motivating the systematic study of such a preorder to understand how different tensors relate to one another. This thesis addresses the problem of constructing and comparing the restriction preorder for a class of order-3 tensors whose support can be interpreted as partial Latin squares. We first classify all 3 × 3 × 3 partial Latin square tensors of weight up to five into main classes, which are equivalence classes under slice and mode permutations. We then generalize the notion of restriction to allow for mode permutations, which we call extended restriction. To determine whether an extended restriction exists between two tensors, we translate the restriction equations---one for each possible mode permutation of the target tensor---into a system of multivariate polynomials. The existence of a common solution to this system, which we test using Gröbner bases, decides the restriction. By applying this method to the classified tensors, we construct the extended restriction preorder between main classes and visualize it as a Hasse diagram. The results demonstrate that while the Gröbner basis approach is exact, it is currently limited on consumer hardware to problems with a combined tensor weight not exceeding eight. This work establishes a framework for the exact, symbolic computation of tensor restrictions and provides a foundation for future explorations of larger and more complex tensors.