Embedding properties of endomorphism semigroups

João Araújo ; Friedrich Wehrung

Fundamenta Mathematicae, Tome 205 (2009), p. 125-146 / Harvested from The Polish Digital Mathematics Library

Résumé

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 infinite-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 fewer than $2^{card Ω}$ operations, then End F has no semigroup embedding into its dual. The bound $2^{card Ω}$ is optimal. (4) Let F be a free left module over a left ℵ₁-noetherian 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 G. M. Bergman and B. M. Schein. We also formalize our results in the setting of algebras endowed with a notion of independence (in particular, independence algebras).

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Publié le : 2009-01-01

Zbl 1175.20053

EUDML-ID : urn:eudml:doc:286492