Activating a sequence of the binary variables in a multi-dimensional array

Question

Suppose we have an array of binary variables $\{x_{i,t,n}, \ \forall i \in I, t \in T, n \in N \}$. If we want to define a condition as, if any of $x_{i,t,n} = 1$ in an arbitrary index $i$, then the rest of the bigger index $i$ should be one. (E.g. if the card of index I would be six, then one of the possible solutions would be $\{0,0,1,1,1,1\}$. (For simplicity, I omitted the sets $T$ and $N$). I know there is a constraint as $x_{i,t,n} \leq x_{i+1,t,n}$, but as far as I test in some problems, it sorts the variables from the first position. For example in the scheduling problem when we want to deploy the resources respectively. Actually, I am not sure it can be used in general at all. I tried to write that in the following form, and I would like to know other insights regarding that.

$$ ((x_{i} \implies(x_{j} \land x_{k})) \bigvee (x_{j} \implies(\lnot x_{i} \land x_{k})) \bigvee (x_{k} \implies(\lnot x_{i} \land \lnot x_{j}))) \quad \forall i,j,k \in I: (i\lt j \lt k) $$

Answer 1

Introduce an ordering $\text{ord}: I \to \{1,\dots,|I|\}$ and impose $$x_i \le x_j$$ for all $i\in I$ and $j\in I$ such that $\text{ord}_i + 1 = \text{ord}_j$.

An alternative, less efficient, approach with $O(|I|^2)$ constraints is $$x_i \le x_j$$ for all $i\in I$ and $j\in I$ such that $\text{ord}_i < \text{ord}_j$.