Embedding properties of endomorphism semigroups

João Araújo, Friedrich Wehrung
2009 Fundamenta Mathematicae  
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 Self Γ iff card Γ 2 card Ω . 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
more » ... -dimensional, then there is no embedding from (Sub V, +) into (Sub V, ∩) and no embedding from (End V, •) into its dual semigroup. (3) Let F be an algebra freely generated by an infinite subset Ω. If F has less than 2 card Ω operations, then End F has no semigroup embedding into its dual. The cardinality bound 2 card Ω is optimal. (4) Let F be a free left module over a left ℵ 1 -noetherian ring (i.e., a ring without strictly increasing chains, of length ℵ 1 , 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 endowed with a notion of independence (in particular independence algebras).
doi:10.4064/fm202-2-2 fatcat:jm7yw446ibcshcel6nvfylm4xe