  In the paper we study the preservation of pseudocompactness (resp., countable
compactness, sequential compactness, $\omega$-boundedness, totally countable
compactness, countable pracompactness, sequential pseudocompactness) by
Tychonoff products of pseudocompact (and countably compact) to\-pological
Brandt $\lambda_i^0$-extensions of semitopological monoids with zero. In
particular we show that if $\big\{
\big(B^0_{\lambda_i}(S_i),\tau^0_{B(S_i)}\big) \colon i\in\mathscr{I}\big\}$ is
a family of Hausdorff pseudocompact to\-pological Brandt
$\lambda_i^0$-extensions of pseudocompact semitopological monoids with zero
such that the Tychonoff product $\prod\left\{ S_i \colon
i\in\mathscr{I}\right\}$ is a pseudocompact space then the direct product
$\prod\big\{ \big(B^0_{\lambda_i}(S_i),\tau^0_{B(S_i)}\big) \colon
i\in\mathscr{I}\big\}$ endowed with the Tychonoff topology is a Hausdorff
pseudocompact semitopological semigroup.
