Clarification on the formalization of tautological consequence

Question

I have a mediocre question.

Just to clarify, if a is a tautological consequence of b, then a ∨ b = b?

It is assumed that a and b are literals.

Answer 1

If $a$ and $b$ are distinct literals, then $a$ cannot be a tautological consequence of $b$!

And if $a$ and $b$ are literals, '$a \lor b = b$' is ill formed.

If $\alpha$ and $\beta$ are wffs more generally and $\alpha$ is a tautological consequence of $\beta$, then $\beta \to (\alpha \lor \beta)$ will be a tautology. But $(\alpha \lor \beta) \to \beta$ won't in general be a tautology. So we won't in general have $(\alpha \lor \beta) \equiv \beta$ a tautology -- i.e. we won't have $\vDash (\alpha \lor \beta) \equiv \beta$.