It is certainly possible to construct a 'number system' in which $\dots999 = -1$ is true...

Define $\mathbf H = \{n:\mathbb W \to \mathbb Z\}$ to be the set of all infinite sequences of integers and represent the the sequence $n_0, n_1, n_2, \dots$ as $(\dots, n_2, n_1, n_0)$. these sequences will, formally, correspond to the sequences $n_0 + 10n_1 + 100n_2 + \dots$.

Define addition by $(m+n)_i = m_i + n_i$ and multiplication by $(m n)_i = \sum_{k=0}^i m_k n_{i-k}$. Other operation could be define similarly.

Say that $n \in \mathbf H$ is in canonical form if, for every $i \in \mathbb W$, $0 \le n_i \le 9$ or for every $i \in \mathbb W$, $-9 \le n_i \le 0$.

You need to define which sequences are null sequences, that is, sequences that correspond to the integer $0$. For example $(\dots, 0, -1, 10)$ is a null sequence. Define two sequences to be congruent if their difference is an null sequence. Prove that congruence is an equivalence relation.

Finally, define $\mathbf H_{10}$ to be the corresponding set of equivalence classes. The set of all sequences in canonical form is a transversal of $\mathbf H_{10}$.

In $\mathbf H_{10}$, it is true that $(\cdots,9,9,9) = (\cdots,0,0,-1).$

The problem is that $\mathbb Z$ is a proper subset of $\mathbf H_{10}$, that is, $\mathbf H_{10}$ is bigger than $\mathbb Z$ and $\dots999$ is in $\mathbf H_{10}$ but it is not in $\mathbb Z$.

It is certainly possible to construct a 'number system' in which $\dots999 = -1$ is true...

Define $\mathbf H = \{n:\mathbb W \to \mathbb Z\}$ to be the set of all infinite sequences of integers and represent the the sequence $n_0, n_1, n_2, \dots$ as $(\dots, n_2, n_1, n_0)$. these sequences will, formally, correspond to the sequences $n_0 + 10n_1 + 100n_2 + \dots$.

Define addition by $(m+n)_i = m_i + n_i$ and multiplication by $(m n)_i = \sum_{k=0}^i m_k n_{i-k}$. Other operation could be define similarly.

Say that $n \in \mathbf H$ is in canonical form if, for every $i \in \mathbb W$, $0 \le n_i \le 9$ or for every $i \in \mathbb W$, $-9 \le n_i \le 0$.

You need to define which sequences are null sequences, that is, sequences that correspond to the integer $0$. For example $(\dots, 0, -1, 10)$ is a null sequence. Say, $n \in \mathbf H$ is a null sequence if, for every $\alpha \in \mathbb W$, there exists a $\beta \in \mathbb W$ such that $\beta > \alpha$ and $\sum_{i=0}^\beta 10^in_i = 0.$ Define two sequences to be congruent if their difference is a null sequence. Prove that congruence is an equivalence relation.

Finally, define $\mathbf H_{10}$ to be the corresponding set of equivalence classes. The set of all sequences in canonical form is a transversal of $\mathbf H_{10}$.

In $\mathbf H_{10}$, it is true that $(\cdots,9,9,9) = (\cdots,0,0,-1).$

The problem is that $\mathbb Z$ is a proper subset of $\mathbf H_{10}$, that is, $\mathbf H_{10}$ is bigger than $\mathbb Z$ and $\dots999$ is in $\mathbf H_{10}$ but it is not in $\mathbb Z$.