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## Embedding properties of endomorphism semigroups

### Volume 202 / 2009

Fundamenta Mathematicae 202 (2009), 125-146 MSC: Primary 20M20; Secondary 08A35, 08A05, 15A03, 05B35. DOI: 10.4064/fm202-2-2

#### Abstract

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).

#### Authors

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
e-mail
• Friedrich WehrungLMNO, CNRS UMR 6139
Université de Caen
Campus 2,
Département de Mathématiques
BP 5186
14032 Caen Cedex, France
e-mail

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