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Faithful linear representations of bands

F. Cedó and J. Okniński

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A band is a semigroup consisting of idempotents. It is proved that for any field $K$ and any band $S$ with finitely many components, the semigroup algebra $K[S]$ can be embedded in upper triangular matrices over a commutative $K$-algebra. The proof of a theorem of Malcev on embeddability of algebras into matrix algebras over a field is corrected and it is proved that if $S=F\cup E$ is a band with two components $E$, $F$ such that $F$ is an ideal of $S$ and $E$ is finite, then $S$ is a linear semigroup. Certain sufficient conditions for linearity of a band $S$, expressed in terms of annihilators associated to $S$, are also obtained.

Article information

Publ. Mat., Volume 53, Number 1 (2009), 119-140.

First available in Project Euclid: 17 December 2008

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 20M25: Semigroup rings, multiplicative semigroups of rings [See also 16S36, 16Y60] 16R20: Semiprime p.i. rings, rings embeddable in matrices over commutative rings 16S36: Ordinary and skew polynomial rings and semigroup rings [See also 20M25]
Secondary: 20M12: Ideal theory 20M17: Regular semigroups 20M30: Representation of semigroups; actions of semigroups on sets

Linear band semigroup algebra triangular matrices annihilator PI rings normal band


Cedó, F.; Okniński, J. Faithful linear representations of bands. Publ. Mat. 53 (2009), no. 1, 119--140.

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