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

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 $$j≽_i k$$.

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 $$≽_i$$.