Pseudocompactness, products and topological Brandt ${\lambda }^0$-extensions of semitopological monoids

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 $\prod\left{ S_i \colon i\in\mathscr{I}\right}$isapseudocompactspacethenthedirectproduct$\prod\big{ \big(B^0_{\lambda_i}(S_i),\tau^0_{B(S_i)}\big) \colon i\in\mathscr{I}\big}$ endowed with the Tychonoff topology is a Hausdorff pseudocompact semitopological semigroup.