A textbook says in Heyting Algebra,
The pseudo-complement of an element $a$ is denoted as $a^{\ast}$. One of the DeMorgan's law $\left(\vee a_{i}\right)^{\ast}=\wedge a_{i}^{\ast}$ holds whenever $\vee a_{i}$ makes sense. Note that the other DeMorgan's law, i.e. $\left(\wedge a_{i}\right)^{\ast}=\vee a_{i}^{\ast}$ does not hold in general.
Firstly, is this correct?
If I consider the duality, in co-Heyting Algebra,
The pseudo-complement of an element $a$ is denoted as $a^{\ast}$. One of the DeMorgan's law $\left(\vee a_{i}\right)^{\ast}=\wedge a_{i}^{\ast}$ does not hold in general. But the other DeMorgan's law, i.e. $\left(\wedge a_{i}\right)^{\ast}=\vee a_{i}^{\ast}$ holds.
Secondary, is this correct?
Finally, if we consider bi-Heyting Algebra, Do both of the DeMorgan's laws hold?