Embedding properties of endomorphism semigroups

Abstract

Denote by PSelf Ω (resp., Self Ω) the partial (resp., full) transformation monoid over a set Ω, and by Sub V (resp., End V ) the collection of all subspaces (resp., endomorphisms) of a vector space V . We prove various results that imply the following: (1) If card Ω 􏰆 2, then Self Ω has a semigroup embedding into the dual of SelfΓ iff cardΓ 􏰆 2cardΩ. In particular, if Ω has at least two elements, then there exists no semigroup embedding from Self Ω into the dual of PSelf Ω. (2) If V is infinite-dimensional, then there is no embedding from (Sub V, +) into (SubV,∩) and no embedding from (EndV,◦) into its dual semigroup. (3) Let F be an algebra freely generated by an infinite subset Ω. If F has less than 2card Ω operations, then End F has no semigroup em- bedding into its dual. The cardinality bound 2card Ω is optimal. (4) Let F be a free left module over a left א1-nœtherian ring (i.e., a ring without strictly increasing chains, of length א1, of left ideals). Then End F has no semigroup embedding into its dual. (1) and (2) above solve questions proposed by B. M. Schein and G. M. Bergman. We also formalize our results in the settings of algebras en- dowed with a notion of independence (in particular independence alge- bras).