Monadic MV-algebras II: Monadic implicational subreducts

In this paper, we study the class of all monadic implicational subreducts, that is, the {→,∀,1}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\{\rightarrow, \forall,1\}}$$\end{document}-subreducts of the class of monadic MV-algebras. We prove that this class is an equational class, which we denote by ML\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{ML}}$$\end{document}, and we give an equational basis for this variety. An algebra in ML\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{ML}}$$\end{document} is called a monadic Łukasiewicz implication algebra. We characterize the subdirectly irreducible members of ML\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{ML}}$$\end{document} and the congruences of every monadic Łukasiewicz implication algebra by monadic filters. We prove that ML\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{ML}}$$\end{document} is generated by its finite members. Finally, we completely describe the lattice of subvarieties, and we give an equational basis for each proper subvariety.