(1) Is the product of lexsegment ideals again a lexsegment ideal?
(2) Is the product of (strongly) stable ideals again (strongly) stable?
I know that both of them are false and I can find examples for (1) but I am not able to find examples for (2). Would it be possible for you to help me to find examples for (2)? (Jürgen Herzog-Takayuki Hibi, Monomial Ideals, problem 6.8.)
If you have the answer for products of lexsegment ideals, then it is easy to try a similar thing for (strongly) stable ideals, since lexsegment ideals are (strongly) stable, and sometimes their product is not even stable anymore. If $I=(x_1,x_2)$ and $J=(x_2,x_3)$, then $IJ=(x_1x_2,x_1x_3,x_2^2,x_2x_3)$ in $K[x_1,x_2,x_3]$. Now $x_2$ divides $w=x_1x_2$, but $wx_1/x_2=x_1^2$ is not in $IJ$.
Another remark: stable (monomial) ideals have linear quotients. In the ring $K[x_1,x_2,x_3,x_4]$ the ideals $I=(x_3,x_4)$ and $J=(x_1^2x_2,x_1x_2x_3,x_2x_3x_4,x_3x_4^2)$ have linear quotients, but the product $IJ$ has not (see Conca-Herzog paper). So also the property of having linear quotients is not inherited by products.
On the other hand, the product of polymatroidal ideals is again polymatroidal (Herzog and Hibi, Cohen-Macaulay polymatroidal ideals).