Universality of the lattice of transformation monoids (Q2845449)

The authors solve an open problem in [\textit{M. Goldstern} and \textit{M. Pinsker}, Algebra Univers. 59, No. 3--4, 365--403 (2008; Zbl 1201.08003)]:NEWLINENEWLINENEWLINETheorem 1.1: Mon(\(\lambda\)) is \textit{universal} for complete algebraic lattices with at most \(2^{\lambda}\) compact elements with respect to closed sublattices; i.e., the closed sublattices on Mon(\(\lambda\)) are, up to isomorphism, precisely the complete algebraic lattices with at most \(2^{\lambda}\) compact elements.NEWLINENEWLINEAs a corollary, they also prove that if \(L\) is an algebraic lattice with at most \(2^{\lambda}\) compact elements, then it is even isomorphic to a closed sublattice of Mon(\(\lambda\)) via an isomorphism which preserves the smallest element.NEWLINENEWLINELet \(\lambda\) be a fixed infinite set of cardinality \(\lambda\) (the set is identified with its cardinality). A subset of \(\lambda^{\lambda}\) which is closed under composition and contains the identity function is called a transformation monoid on \(\lambda\) and denoted by Mon(\(\lambda\)).