  Consider a matroid equipped with a labeling of its ground set to an abelian
group. We define the label of a subset of the ground set as the sum of the
labels of its elements. We study a collection of problems on finding bases and
common bases of matroids with restrictions on their labels. For zero bases and
zero common bases, the results are mostly negative. While finding a non-zero
basis of a matroid is not difficult, it turns out that the complexity of
finding a non-zero common basis depends on the group. Namely, we show that the
problem is hard for a fixed group if it contains an element of order two,
otherwise it is polynomially solvable.
  As a generalization of both zero and non-zero constraints, we further study
$F$-avoiding constraints where we seek a basis or common basis whose label is
not in a given set $F$ of forbidden labels. Using algebraic techniques, we give
a randomized algorithm for finding an $F$-avoiding common basis of two matroids
represented over the same field for finite groups given as operation tables.
The study of $F$-avoiding bases with groups given as oracles leads to a
conjecture stating that whenever an $F$-avoiding basis exists, an $F$-avoiding
basis can be obtained from an arbitrary basis by exchanging at most $|F|$
elements. We prove the conjecture for the special cases when $|F|\le 2$ or the
group is ordered. By relying on structural observations on matroids
representable over fixed, finite fields, we verify a relaxed version of the
conjecture for these matroids. As a consequence, we obtain a polynomial-time
algorithm in these special cases for finding an $F$-avoiding basis when $|F|$
is fixed.
