# Existence of infimum on the set of equivalence classes

Let $$Q$$ be a complete lattice.

Let $$\sim$$ be an equivalence relation on $$Q$$ conforming to the axiom $$(f_0\sim f_1\wedge g_0\sim g_1\wedge f_0\leq g_0)\Rightarrow f_1\leq g_1.$$

Define the order on the set $$Q/\sim$$ of equivalence classes in the obvious way.

Now let $$S$$ be a (nonempty) set of equivalence classes. Is it true that the infimum $$\inf S$$ necessarily exists and $$\inf S=[\inf_{X\in S} A_X]$$ where $$A_X\in X$$ for every equivalence class $$X$$?

$$[\inf_{X\in S}A_X]\leq X$$ for every $$X\in S$$ because $$\inf_{X\in S}A_X\leq A_X$$.
Suppose an equivalence class $$L\leq X$$ for every $$X\in S$$.
Take $$l\in L$$. We have $$l\leq A_X$$; $$l\leq\inf_{X\in S}A_X$$; $$L\leq[\inf_{X\in S}A_X]$$.
Thus $$[\inf_{X\in S}A_X]$$ is the greatest lower bound of $$S$$.