TY - JOUR
T1 - Representation theory of finite semigroups, semigroup radicals and formal language theory
AU - Almeida, Jorge
AU - Margolis, Stuart
AU - Steinberg, Benjamin
AU - Volkov, Mikhail
PY - 2009/3
Y1 - 2009/3
N2 - In this paper we characterize the congruence associated to the direct sum of all irreducible representations of a finite semigroup over an arbitrary field, generalizing results of Rhodes for the field of complex numbers. Applications are given to obtain many new results, as well as easier proofs of several results in the literature, involving: triangularizability of finite semigroups; which semigroups have (split) basic semigroup algebras, two-sided semidirect product decompositions of finite monoids; unambiguous products of rational languages; products of rational languages with counter; and Černý's conjecture for an important class of automata.
AB - In this paper we characterize the congruence associated to the direct sum of all irreducible representations of a finite semigroup over an arbitrary field, generalizing results of Rhodes for the field of complex numbers. Applications are given to obtain many new results, as well as easier proofs of several results in the literature, involving: triangularizability of finite semigroups; which semigroups have (split) basic semigroup algebras, two-sided semidirect product decompositions of finite monoids; unambiguous products of rational languages; products of rational languages with counter; and Černý's conjecture for an important class of automata.
KW - Language theory
KW - Radicals
KW - Representation theory
UR - http://www.scopus.com/inward/record.url?scp=69049118047&partnerID=8YFLogxK
U2 - 10.1090/s0002-9947-08-04712-0
DO - 10.1090/s0002-9947-08-04712-0
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AN - SCOPUS:69049118047
SN - 0002-9947
VL - 361
SP - 1429
EP - 1461
JO - Transactions of the American Mathematical Society
JF - Transactions of the American Mathematical Society
IS - 3
ER -