Existence of infimum on the set of equivalence classes

Question

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 $X$?

Answer 1

$[\inf_{X\in S}A_X]\leq X$ 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 $S$.