Domains whose ideals meet a universal restriction

Let $S(D)$ represent a set of proper nonzero ideals $I(D)$ (resp., $t$ -ideals $I_{t}(D)$) of an integral domain $D\neq qf(D)$ and let $P$ be a valid property of ideals of $D.$ We say $S(D)$ meets $P$ (denoted $S(D)\vartriangleleft P)$ if each $s\in S(D)$ is contained in an ideal satisfying $P$. If $S(D)$ $\vartriangleleft P,$ $\dim (D)$ can't be controlled. When $R=D[X],$ $I(D)$ $\vartriangleleft P$ does not imply $I(R)$ $\vartriangleleft P$ while $I_{t}(D)$ $\vartriangleleft P$ implies $I_{t}(R)$ $\vartriangleleft P$ usually. We say $S(D)$ meets $P$ with a twist $($ written $S(D)\vartriangleleft ^{t}P)$ if each $s\in S(D)$ is such that, for some $n\in N,$ $s^{n}$ is contained in an ideal satisfying $P$ and study $S(D)\vartriangleleft ^{t}P,$ as its predecessor. A modification of the above approach is used to give generalizations of Almost Bezout domains.