Classes of Matroids Closed Under Minors and Principal Extensions

This work studies the classes of matroids that are closed under minors, addition of coloops and principal extensions. To any matroid M in such a class a matroid M ° is constructed such that it contains M as a minor, has all proper minors in the class and violates Zhang- Yeung inequality. When the class enjoys the inequality the matroid M ° becomes an excluded minor. An analogous assertion was known before for the linear matroids over any infinite field in connection with Ingleton inequality. The result is applied to the classes of multilinear, algebraic and almost entropic matroids. In particular, the class of almost entropic matroids has infinitely many excluded minors.