Embedding properties of endomorphism semigroups

João Araújo¹ ; Friedrich Wehrung²

¹ Universidade Aberta Rua da Escola Politécnica, 147 1269-001 Lisboa, Portugal and Centro de Álgebra da Universidade de Lisboa Av. Gama Pinto, 2 1649-003 Lisboa, Portugal
² LMNO, CNRS UMR 6139 Université de Caen Campus 2, Département de Mathématiques BP 5186 14032 Caen Cedex, France

Fundamenta Mathematicae, Tome 202 (2009) no. 2, pp. 125-146.

Résumé

Denote by $\mathop{\rm PSelf}\nolimits\varOmega$ (resp., $\mathop{\rm Self}\nolimits\varOmega$) the partial (resp., full) transformation monoid over a set $\varOmega$, and by $\mathop{\rm Sub}\nolimits V$ (resp., $\mathop{\rm End}\nolimits V$) the collection of all subspaces (resp., endomorphisms) of a vector space $V$. We prove various results that imply the following:(1) If $\mathop{\rm card}\nolimits\varOmega\ge2$, then $\mathop{\rm Self}\nolimits\varOmega$ has a semigroup embedding into the dual of $\mathop{\rm Self}\nolimits\varGamma$ iff $\mathop{\rm card}\nolimits\varGamma\ge2^{\mathop{\rm card}\nolimits\varOmega}$. In particular, if $\varOmega$ has at least two elements, then there exists no semigroup embedding from $\mathop{\rm Self}\nolimits\varOmega$ into the dual of $\mathop{\rm PSelf}\nolimits\varOmega$. (2) If $V$ is infinite-dimensional, then there is no embedding from $(\mathop{\rm Sub}\nolimits V,+)$ into $(\mathop{\rm Sub}\nolimits V,\cap)$ and no embedding from $(\mathop{\rm End}\nolimits V,\circ)$ into its dual semigroup. (3) Let $F$ be an algebra freely generated by an infinite subset $\varOmega$. If $F$ has fewer than $2^{\mathop{\rm card}\nolimits\varOmega}$ operations, then $\mathop{\rm End}\nolimits F$ has no semigroup embedding into its dual. The bound $2^{\mathop{\rm card}\nolimits\varOmega}$ is optimal. (4) Let $F$ be a free left module over a left $\aleph_1$-noetherian ring (i.e., a ring without strictly increasing chains, of length $\aleph_1$, of left ideals). Then $\mathop{\rm End}\nolimits F$ 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).

Détail

DOI : 10.4064/fm202-2-2

Keywords: denote mathop pself nolimits varomega resp mathop self nolimits varomega partial resp full transformation monoid set nbsp varomega mathop sub nolimits resp mathop end nolimits collection subspaces resp endomorphisms vector space nbsp prove various results imply following mathop card nolimits varomega mathop self nolimits varomega has semigroup embedding dual mathop self nolimits vargamma mathop card nolimits vargamma mathop card nolimits varomega particular varomega has least elements there exists semigroup embedding nbsp mathop self nolimits varomega dual nbsp mathop pself nolimits varomega nbsp infinite dimensional there embedding mathop sub nolimits mathop sub nolimits cap embedding mathop end nolimits circ its dual semigroup nbsp algebra freely generated infinite subset nbsp varomega nbsp has fewer nbsp mathop card nolimits varomega operations nbsp mathop end nolimits has semigroup embedding its dual bound nbsp mathop card nolimits varomega optimal module aleph noetherian ring ring without strictly increasing chains length nbsp aleph ideals mathop end nolimits has semigroup embedding its dual above solve questions proposed nbsp nbsp bergman nbsp nbsp schein formalize results setting algebras endowed notion independence particular independence algebras