2008/01/17 by João Araüjo, Joao Araujo, Araujo, Joao +2
Computer Science · Mathematics · #15A03 #16S50 (Secondary) #20M30 (Primary) #Advanced Algebra and Logic #FOS: Mathematics #Rings and Algebras (math.RA) #Rings, Modules, and Algebras #math.RA #msc:15A03 #msc:16S50 #msc:20M30 #semigroups and automata theory
paper · pdf · doi:10.48550/arxiv.0801.2644
To appear in Fundamenta Mathematicae
openalex publication_date 2008/01/17 · arxiv created 2008/10/15 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/29
Denote by PSelf(X) (resp., Self(X)) the partial (resp., full) transformation monoid over a set X, 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 X has at least two elements, then Self(X) has a semigroup embedding into the dual of Self(Y) iff card(Y) >= 2card(X). In particular, if X has at least two elements, then there exists no semigroup embedding from Self(X) into the dual of PSelf(X). (2) If V is infinite-dimensional, then there are no embedding from (Sub(V),+) into (Sub(V),∩) and no semigroup embedding from End(V) into its dual. (3) Let F be an algebra freely generated by an infinite subset X. If F has less than 2card(X) operations, then End(F) has no semigroup embedding into its dual. The cardinality bound 2card(X) is optimal. (4) Let F be a free left module over a left aleph one - noetherian ring (i.e., a ring without strictly increasing chains, of length aleph one, 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).