Consider the following definition of matroid.
A matroid over a set $X$ is a family $\mathcal B\subseteq\mathcal P(X)$ of subsets of $X$ (the set of bases) with the following properties:
- $\mathcal B$ is non-empty.
- (base exchange property) For any two $A,B\in \mathcal B$ and $a\in A\setminus B$, there is a $b\in B\setminus A$, so that $$A\setminus\{a\}\cup\{b\}\in \mathcal B\qquad\text{and}\qquad B\setminus\{b\}\cup\{a\}\in \mathcal B $$
I am interested in a generalization, lets call it $k$-matroids, in which I replace the second axiom by the following:
- For any two $A,B\in \mathcal B$ and distinct $a_1,...,a_k\in A\setminus B$, there is are distinct $b_1,...,b_k\in B\setminus A$, so that $$A\setminus\{a_1,...,a_k\}\cup\{b_1,...,b_k\}\in \mathcal B\qquad\text{and}\qquad B\setminus\{b_1,...,b_k\}\cup\{a_1,...,a_k\}\in \mathcal B $$
Is this generalization known and if so, where to read about it. If not, are there some obvious connections to classical matroids (which are 1-matroids).
Your second condition still defines a matroid, no? By a repeated application of the standard basis exchange you can obtain your "stronger" k exchange property on all matroids I think:
Apply basis exchange to get another base B' = B1 - a_1 + b_1 for some b_1. Then apply basis exchange on B' and B2 to replace a2 with something from B2 again, etc.