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If, on the other hand, you're actually interested in the case where the elements of your monoid are non-empty, finite families of subsets of $V$ (are you?), then things are quite different and, with the same notation as above, you would better move from the full power monoid $\mathcal P(H)$ to the power monoid $\mathcal P_{\rm fin}(H)$ of $H$, that is, the submonoid of $\mathcal P(H)$ whose elements are the (non-empty) finite subsets of $H$: It is obvious that $\mathcal P_{\rm fin}(H)$ is strongly $\subseteq$-artinian, meaning that the $\subseteq$-height of every $X \in \mathcal P_{\rm fin}(H)$ is finite; as a result, every $X \in \mathcal P_{\rm fin}(H)$ factors into a (finite) product of $\subseteq$-quarks (see the notes below). Then, you are left with the problem of characterizing the $\subseteq$-quarks. (You could also use other preorders, but the inclusion order is a natural go-to here.)

Notes. Given a preorder $\preceq$ on a monoid $M$, a $\preceq$-non-unit of $M$ is an element $u \in M$ that is not $\preceq$-equivalent to the identity $1_M$ (two elements $x, y \in M$ are $\preceq$-equivalent if $x \preceq y \preceq x$). Then, the $\preceq$-height of an element $x \in M$ is the supremum of the set of all integers $n \ge 1$ for which there exist $\preceq$-non-units $x_1, \ldots, x_n \in M$ with $x_1 = x$ and $x_{i+1} \prec x_i$ for $1 \le i < n$, with the understanding that $\sup \emptyset := 0$; and a $\preceq$-quark of $M$ is a $\preceq$-minimal $\preceq$-non-unit.

If, on the other hand, you're actually interested in the case where the elements of your monoid are non-empty, finite families of subsets of $V$ (are you?), then things are quite different and, with the same notation as above, you would better move from the full power monoid $\mathcal P(H)$ to the power monoid $\mathcal P_{\rm fin}(H)$ of $H$, that is, the submonoid of $\mathcal P(H)$ whose elements are the (non-empty) finite subsets of $H$: It is obvious that $\mathcal P_{\rm fin}(H)$ is strongly $\subseteq$-artinian, meaning that the $\subseteq$-height of every $X \in \mathcal P_{\rm fin}(H)$ is finite; as a result, every $X \in \mathcal P_{\rm fin}(H)$ factors into a (finite) product of $\subseteq$-irreducibles (see the notes below). Then, you are left with the problem of characterizing the $\subseteq$-irreducibles. (You could also use other preorders, but the inclusion order is a natural go-to here.)

Notes. Given a preorder $\preceq$ on a monoid $M$, a $\preceq$-non-unit of $M$ is an element $u \in M$ that is not $\preceq$-equivalent to the identity $1_M$ (two elements $x, y \in M$ are $\preceq$-equivalent if $x \preceq y \preceq x$). Then, the $\preceq$-height of an element $x \in M$ is the supremum of the set of all integers $n \ge 1$ for which there exist $\preceq$-non-units $x_1, \ldots, x_n \in M$ with $x_1 = x$ and $x_{i+1} \prec x_i$ for $1 \le i < n$, with the understanding that $\sup \emptyset := 0$; and a $\preceq$-irreducible of $M$ is a $\preceq$-non-unit $a \in M$ such that $a \ne xy$ for all $\preceq$-non-units $x, y \in M$ with $x \prec a$ and $y \prec a$.

If, on the other hand, you're actually interested in the case where the elements of your monoid are non-empty, finite families of subsets of $V$ (are you?), then things are quite different and, with the same notation as above, you would better move from the full power monoid $\mathcal P(H)$ to the power monoid $\mathcal P_{\rm fin}(H)$ of $H$, that is, the submonoid of $\mathcal P(H)$ whose elements are the (non-empty) finite subsets of $H$: It is obvious that $\mathcal P_{\rm fin}(H)$ is strongly $\subseteq$-artinian, meaning that the $\subseteq$-height of every $X \in \mathcal P_{\rm fin}(H)$ is finite; and hence every $X \in \mathcal P_{\rm fin}(H)$ factors into a (finite) product of $\subseteq$-quarks (see the notes below). Then, you are left with the problem of characterizing the $\subseteq$-quarks. (You could also use other preorders, but the inclusion order is a natural choice here.)

Notes. Given a preorder $\preceq$ on a monoid $M$, a $\preceq$-non-unit of $H$ is an element $u \in H$ that is not $\preceq$-equivalent to the identity $1_H$ (two elements $x, y \in H$ are $\preceq$-equivalent if $x \preceq y \preceq x$). Then, the $\preceq$-height of an element $x \in M$ is the supremum of the set of all integers $n \ge 1$ for which there exist $\preceq$-non-units $x_1, \ldots, x_n \in H$ with $x_1 = x$ and $x_{i+1} \prec x_i$ for $1 \le i < n$, with the understanding that $\sup \emptyset := 0$; and a $\preceq$-quark of $H$ is a $\preceq$-minimal $\preceq$-non-unit $a \in H$.

If, on the other hand, you're actually interested in the case where the elements of your monoid are non-empty, finite families of subsets of $V$ (are you?), then things are quite different and, with the same notation as above, you would better move from the full power monoid $\mathcal P(H)$ to the power monoid $\mathcal P_{\rm fin}(H)$ of $H$, that is, the submonoid of $\mathcal P(H)$ whose elements are the (non-empty) finite subsets of $H$: It is obvious that $\mathcal P_{\rm fin}(H)$ is strongly $\subseteq$-artinian, meaning that the $\subseteq$-height of every $X \in \mathcal P_{\rm fin}(H)$ is finite; as a result, every $X \in \mathcal P_{\rm fin}(H)$ factors into a (finite) product of $\subseteq$-quarks (see the notes below). Then, you are left with the problem of characterizing the $\subseteq$-quarks. (You could also use other preorders, but the inclusion order is a natural go-to here.)

Notes. Given a preorder $\preceq$ on a monoid $M$, a $\preceq$-non-unit of $M$ is an element $u \in M$ that is not $\preceq$-equivalent to the identity $1_M$ (two elements $x, y \in M$ are $\preceq$-equivalent if $x \preceq y \preceq x$). Then, the $\preceq$-height of an element $x \in M$ is the supremum of the set of all integers $n \ge 1$ for which there exist $\preceq$-non-units $x_1, \ldots, x_n \in M$ with $x_1 = x$ and $x_{i+1} \prec x_i$ for $1 \le i < n$, with the understanding that $\sup \emptyset := 0$; and a $\preceq$-quark of $M$ is a $\preceq$-minimal $\preceq$-non-unit.

If, on the other hand, you're actually interested in the case where the elements of your monoid are non-empty, finite families of subsets of $V$ (are you?), then things are quite different and, with the same notation as above, you would better move from the full power monoid $\mathcal P(H)$ to the power monoid $\mathcal P_{\rm fin}(H)$ of $H$, that is, the submonoid of $\mathcal P(H)$ whose elements are the (non-empty) finite subsets of $H$: It is obvious that $\mathcal P_{\rm fin}(H)$ is strongly $\subseteq$-artinian, meaning that the $\subseteq$-height of every $X \in \mathcal P_{\rm fin}(H)$ is finite, and hence every $X \in \mathcal P_{\rm fin}(H)$ factors into a (finite) product of $\subseteq$-quarks (see the notes below). Then, you are left with the problem of characterizing the $\subseteq$-quarks.

Notes. Given a preorder $\preceq$ on a monoid $M$, the $\preceq$-height of an element $x \in M$ is the supremum of the set of all integers $n \ge 1$ for which there exist $x_1, \ldots, x_n \in H$ with $x_1 = x$ and $x_{i+1} \prec x_i$ for $1 \le i < n$, with the understanding that $\sup \emptyset := 0$; and a $\preceq$-quark of $H$ is a $\preceq$-minimal $\preceq$-non-unit $a \in H$, where a $\preceq$-non-unit is an element of $H$ that is not $\preceq$-equivalent to the identity $1_H$ (two elements $x, y \in H$ are *$\preceq$-equivalent if $x \preceq y \preceq x$).

If, on the other hand, you're actually interested in the case where the elements of your monoid are non-empty, finite families of subsets of $V$ (are you?), then things are quite different and, with the same notation as above, you would better move from the full power monoid $\mathcal P(H)$ to the power monoid $\mathcal P_{\rm fin}(H)$ of $H$, that is, the submonoid of $\mathcal P(H)$ whose elements are the (non-empty) finite subsets of $H$: It is obvious that $\mathcal P_{\rm fin}(H)$ is strongly $\subseteq$-artinian, meaning that the $\subseteq$-height of every $X \in \mathcal P_{\rm fin}(H)$ is finite; and hence every $X \in \mathcal P_{\rm fin}(H)$ factors into a (finite) product of $\subseteq$-quarks (see the notes below). Then, you are left with the problem of characterizing the $\subseteq$-quarks. (You could also use other preorders, but the inclusion order is a natural choice here.)

Notes. Given a preorder $\preceq$ on a monoid $M$, a $\preceq$-non-unit of $H$ is an element $u \in H$ that is not $\preceq$-equivalent to the identity $1_H$ (two elements $x, y \in H$ are $\preceq$-equivalent if $x \preceq y \preceq x$). Then, the $\preceq$-height of an element $x \in M$ is the supremum of the set of all integers $n \ge 1$ for which there exist $\preceq$-non-units $x_1, \ldots, x_n \in H$ with $x_1 = x$ and $x_{i+1} \prec x_i$ for $1 \le i < n$, with the understanding that $\sup \emptyset := 0$; and a $\preceq$-quark of $H$ is a $\preceq$-minimal $\preceq$-non-unit $a \in H$.