A subfamily $\mathcal{G}\subseteq \mathcal{F}\subseteq 2^{[n]}$ of sets is a non-induced (weak) copy of a poset P in $\mathcal{F}$ if there exists a bijection $i:P\to \mathcal{G}$ such that $p{\le }_{P}q$ implies $i(p)\subseteq i(q)$. In the case where in addition $p{\le }_{P}q$ holds if and only if $i(p)\subseteq i(q)$, then $\mathcal{G}$ is an induced (strong) copy of P in $\mathcal{F}$. We consider the minimum number $\mathrm{sat}(n,P)$ [resp. ${\mathrm{sat}}^{⁎}(n,P)$] of sets that a family $\mathcal{F}\subseteq 2^{[n]}$ can have without containing a non-induced [induced] copy of P and being maximal with respect to this property, i.e., the addition of any $G\in 2^{[n]}\setminus \mathcal{F}$ creates a non-induced [induced] copy of P.

We prove for any finite poset P that $\mathrm{sat}(n,P)\le 2^{|P|-2}$, a bound independent of the size n of the ground set. For induced copies of P, there is a dichotomy: for any poset P either ${\mathrm{sat}}^{⁎}(n,P)\le K_P$ for some constant depending only on P or ${\mathrm{sat}}^{⁎}(n,P)\ge {\log }_{2}n$. We classify several posets according to this dichotomy, and also show better upper and lower bounds on $\mathrm{sat}(n,P)$ and ${\mathrm{sat}}^{⁎}(n,P)$ for specific classes of posets.

Our main new tool is a special ordering of the sets based on the colexicographic order. It turns out that if P is given, processing the sets in this order and adding the sets greedily into our family whenever this does not ruin non-induced [induced] P-freeness, we tend to get a small size non-induced [induced] P-saturating family.

一个亚科 $\mathcal{G}\subseteq \mathcal{F}\subseteq 2^{[n]}$集合是一个非诱导（弱）一个偏序集的副本P在$\mathcal{F}$ 如果存在双射 $\mathrm{一世}：磷\to \mathcal{G}$ 以至于 $磷{\le }_{磷}q$ 暗示 $\mathrm{一世}(磷)\subseteq \mathrm{一世}(q)$. 在另外的情况下$磷{\le }_{磷}q$ 成立当且仅当 $\mathrm{一世}(磷)\subseteq \mathrm{一世}(q)$， 然后 $\mathcal{G}$是诱导的（强）的副本P在$\mathcal{F}$. 我们考虑最小数量$\mathrm{坐}(n,磷)$ [分别。 ${\mathrm{坐}}^{⁎}(n,磷)$] 套那一个家庭 $\mathcal{F}\subseteq 2^{[n]}$可以在不包含P的非诱导 [诱导] 副本的情况下具有并且对于该属性是最大的，即，添加任何$G\in 2^{[n]}\setminus \mathcal{F}$创建P的非诱导 [诱导] 副本。

我们证明了任何有限偏序集P是$\mathrm{坐}(n,磷)\le 2^{|磷|-2}$，一个与地面集的大小n无关的边界。对于诱导副本P，有一个二分法：对任何偏序集P要么${\mathrm{坐}}^{⁎}(n,磷)\le {钾}_{磷}$对于一些仅取决于P或的常数${\mathrm{坐}}^{⁎}(n,磷)\ge {\mathrm{日志}}_{2}n$. 我们根据这种二分法对几个偏序集进行分类，并在$\mathrm{坐}(n,磷)$ 和 ${\mathrm{坐}}^{⁎}(n,磷)$ 对于特定类别的poset。

我们的主要新工具是基于字典顺序对集合进行特殊排序。事实证明，如果给定P，按此顺序处理集合并贪婪地将集合添加到我们的家庭中，只要这不会破坏非诱导 [诱导] P自由度，我们往往会得到一个小尺寸的非诱导 [诱导] ] P-饱和族。