Embedding properties of endomorphism semigroups

Tom 202 / 2009

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


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).


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

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