A quasi-additive property of homological shift ideals

In this paper, we investigate which classes of monomial ideals have a quasi-additive property of homological shift ideals. More precisely, for a monomial ideal $I$ we are interested to find out whether $H{S}_{i+j}(I)\subseteq H{S}_i(H{S}_j(I))$. It turns out that $\mathbf{c}$-bounded principal Borel ideals as well as polymatroidal ideals satisfying strong exchange property, and polymatroidal ideals generated in degree two have this quasi-additive property. For squarefree Borel ideals, we even have equality. Besides, the inclusion holds for every equigenerated Borel ideal and polymatroidal ideal when $j=1$.