A note on approximate biprojectivity of some semigroup algebras

We characterize approximate biprojectivity of semigroup algebras related to a Clifford semigroup. As an application, we improve (Corollary 2.6 in Essmaili et al., Arch Math (Basel) 97(2):167–177, 2011) and we characterize pseudo-contractibility of the Clifford semigroup algebra. Namely, we show that ℓ1(S)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \ell ^{1}(S) $$\end{document} is pseudo-contractible if and only if the idempotent set E(S) is locally finite, each maximal subgroup Gp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ G_{p} $$\end{document} is finite for every p∈E(S)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ p\in E(S) $$\end{document}, and ℓ1(S)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \ell ^{1}(S) $$\end{document} has a central approximate identity.