Narrow Results

Given a monomial ideal in a polynomial ring over a field, we define the generalized Newton complementary dual of the given ideal. We show good properties of such duals including linear quotients and isomorphism between the special fiber rings. We construct the cellular free resolutions of duals of strongly stable ideals generated in the same degree. When the base ideal is generated in degree two, we provide an explicit description of cellular free resolution of the dual of a compatible generalized stable ideal.

In this paper, we introduce the notion of extended of an ideal associated to a subset of an $MV$-algebra $A$ and investigate the related properties. A characterization of this extended ideal is given. We show that $I$ is stable ideal relative to $B\subseteq A$ such that $B\cap I=\varnothing$ if and only if $A/I$ is chain $MV$-algebra. Finally, the class $S(B)$ of all stable ideals relative to $B\subseteq A$ is also a complete Heyting algebra, for an $MV$-algebra $A$.