For \(\sigma ,\kappa \) infinite cardinals with \(\kappa \ge \sigma \) , we introduce the notion of \(\sigma \) -matroids on \(\kappa \) . For \(\sigma :=\omega \) , the first infinite cardinal, the notion of \(\sigma \) -matroid coincides with the notion of finitary matroid. One method for obtaining \(\sigma \) -structures is by adding special subsets of \(\kappa \) of size \({\ge }\sigma \) to the collection of independent sets of some already existing matroid structure which contains independent sets only of size \({<}\sigma \) . We show that under some additional order property our newly obtained structure actually forms a matroid. Also we show that we can get \(\sigma \) -structures from any arbitrary matroid structure by considering its circuits. We give two characterizations of some matroid to be a \(\sigma \) -matroid, one in terms of circuit sizes and another by considering certain topology on \(2^\kappa \) . Finally, we briefly discuss some results about matroid unions.
