Universality of the Lattice of Transformation Monoids

The set of all transformation monoids on a fixed set of infinite cardinality λ, equipped with the order of inclusion, forms a complete algebraic lattice Mon(λ) with 2 compact elements. We show that this lattice is universal with respect to closed sublattices, i.e., the closed sublattices of Mon(λ) are, up to isomorphism, precisely the complete algebraic lattices with at most 2 compact elements. 1. Definitions and the result Fix an infinite set – for the sake of simpler notation, we identify the set with its cardinality λ. By a transformation monoid on λ we mean a subset of λ which is closed under composition and which contains the identity function. The set of transformation monoids acting on λ, ordered by inclusion, forms a complete lattice Mon(λ), in which the meet of a set of monoids is simply their intersection. This lattice is algebraic, i.e., every element is a join of compact elements – an element a in a complete lattice L = (L,∨,∧) is called compact iff whenever A ⊆ L and a ≤ ∨ A, then there is a finite A ⊆ A such that a ≤ ∨ A. In the case of Mon(λ), the compact elements are precisely the finitely generated monoids, i.e., those monoids which contain a finite set of functions such that every function of the monoid can be composed from functions of this finite set. Consequently, the number of compact elements of Mon(λ) equals 2. It is well-known and not hard to see that for any cardinal κ, the algebraic lattices with at most κ compact elements are, up to isomorphism, precisely the subalgebra lattices of algebras whose domains have κ elements ([BF48]; see also Theorem 48 in the textbook [Grä03]). For example, Mon(λ) is the subalgebra lattice of the algebra which has domain λ, a binary operation which is the function composition on λ, as well as a constant operation whose value is the identity function on λ. 2010 Mathematics Subject Classification. Primary 06B15; secondary 06B23; 20M20.