Persistence Interval

Next, we want to parametrize the isomorphism classes of the F[t]-modules by suitable objects.

A \mathcal{P}-interval is an ordered pair (i,j) with 0\leq i<j\in\mathbb{Z}^\infty=\mathbb{Z}\cup\{+\infty\}.

We may associate a graded F[t]-module to a set \mathcal{S} of \mathcal{P}-intervals via a bijection Q. We define \displaystyle Q(i,j)=\Sigma^i F[t]/(t^{j-i}) for a \mathcal{P}-interval (i,j). When j=+\infty, we have Q(i,+\infty)=\Sigma^iF[t].

For a set of \mathcal{P}-intervals \mathcal{S}=\{(i_1,j_1),(i_2,j_2),\dots,(i_n,j_n)\}, we define \displaystyle Q(\mathcal{S})=\bigoplus_{k=1}^n Q(i_k, j_k).

We may now restate the correspondence as follows.

The correspondence \mathcal{S}\to Q(\mathcal{S}) defines a bijection between the finite sets of \mathcal{P}-intervals and the finitely generated graded modules over the graded ring F[t].

Hence, the isomorphism classes of persistence modules of finite type over F are in bijective correspondence with the finite sets of \mathcal{P}-intervals.

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