How to mathematically denote that a consumer behaves according to their preference structure at every point in time?

Question

As the title says, I would like to mathematically denote that a consumer behaves according to their preference structure at every time $t$, $t+1$, $t+2$ and so on in a finite consumption set $X$, by consuming their most preferred good in each time period. Assume that it takes 1 unit of time for a consumer to consume 1 good in $X$.

This is what I have - may be wrong/incomplete:

Let $C_{it} (K)$ denote the type of commodity consumer $i$ consumes at time $t$, where $K$ is equal to the set of $n$ commodity types $[1,…,n]$. Consumer $i's$ consumption behaviour will be such that for any pairs $j,k\in K$, $C_{it} (K)=j$, $C_{i,t+h} (K)=k$ if and only if $j≽_i k$.

Answer 1

I'd say let $K_0=K$ and for all $t\ge 0$ define iteratively $K_{t+1}=K_{t}\texttt{\\}\{C_{it}\}$, where $C_{it}\in\arg\max_{C\in K_t} u_i(C)$ and $u_i$ represents $≽_i$.