Wydawnictwa / Czasopisma IMPAN / Fundamenta Mathematicae / Wszystkie zeszyty

Streszczenie

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).