Is $a$ in $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ always the semi-major axis?

Question

In the equation for an ellipse: $$ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 $$ is $a$ always the semi-major axis of the ellipse, even if it is less than $b$? My inference is that it isn't, but $a$ seems to be used somewhat frequently in formulae to denote the semi-major axis specifically.