Embedding properties of endomorphism semigroups

João Araújo, and Friedrich Wehrung. "Embedding properties of endomorphism semigroups." Fundamenta Mathematicae 202.2 (2009): 125-146. <http://eudml.org/doc/286492>.

@article{JoãoAraújo2009,
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 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).},
author = {João Araújo, Friedrich Wehrung},
journal = {Fundamenta Mathematicae},
keywords = {transformations; monoids; semigroups; endomorphisms; vector spaces; subspaces; lattices; C-independent; M-independent; matroids},
language = {eng},
number = {2},
pages = {125-146},
title = {Embedding properties of endomorphism semigroups},
url = {http://eudml.org/doc/286492},
volume = {202},
year = {2009},
}

TY - JOUR
AU - João Araújo
AU - Friedrich Wehrung
TI - Embedding properties of endomorphism semigroups
JO - Fundamenta Mathematicae
PY - 2009
VL - 202
IS - 2
SP - 125
EP - 146
AB - 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).
LA - eng
KW - transformations; monoids; semigroups; endomorphisms; vector spaces; subspaces; lattices; C-independent; M-independent; matroids
UR - http://eudml.org/doc/286492
ER -