On some finitely based representations of semigroups
| Autor: | Nikolay N. Silkin |
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| Rok vydání: | 1998 |
| Předmět: | |
| Zdroj: | Proceedings of the American Mathematical Society. 126:1621-1626 |
| ISSN: | 1088-6826 0002-9939 |
| DOI: | 10.1090/s0002-9939-98-04205-1 |
| Popis: | In this paper we present a method of obtaining finitely based linear representations of possibly infinitely based semigroups. Let R{x1, x2, . . . } be a free associative algebra over a commutative ring R with the countable set of free generators {x1, x2, . . . }. An endomorphism α of R{x1, x2, . . . } is called a semigroup endomorphism if x1α, x2α, . . . are monomials (i.e. finite products of xi’s). An ideal I of R{x1, x2, . . . } is called an S-ideal if I is closed under all semigroup endomorphisms of R{x1, x2, . . . }. Let S be a semigroup, M a faithful module over R. A multiplicative homomorhism ψ : S −→ EndRM is called a (linear) representation of S on M . An element p = p(x1, x2, . . . , xn) of R{x1, x2, . . . } is called an identity of the representation ψ if p(ψ(s1), . . . , ψ(sn)) = 0 for all s1, . . . , sn ∈ S. It is not hard to show that the set of all identities of any representation is an Sideal of R{x1, x2, . . . }. An S-ideal I of R{x1, x2, . . . } is called finitely S-generated if there exist p1, . . . , pk ∈ I such that I is the least S-ideal of R{x1, x2, . . . } containing p1, . . . , pk. A representation ψ is called finitely based if the S-ideal I of its identities is finitely S-generated. Any set {p1, . . . , pk} ⊆ I such that the elements p1, . . . , pk S-generate I is called a finite basis of identities of ψ. S.M.Vovsi and N.H.Shon in [VSh] proved that every representation of a finite group over a field is finitely based. A semigroup version of this problem is open: it is not known whether every representation of a finite semigroup is finitely based. Let A be an associative algebra over R, S = 〈A, ·〉 its multiplicative semigroup, and M be an R-module. Let φ : A −→ EndRM be a homomorphism (R-linear, additive, and multiplicative). The kernel of φ, Kerφ, is a two-sided ideal in A. Define a representation ψ of S on M by mψ(s) = mφ(s) for all m ∈ M , s ∈ S. The representation ψ is a linear representation of S on M . Call the representation ψ associated with φ. From now on we assume that the ring R and the module M have at least one of the following properties: (i) for any integer m there exist r1, . . . , rm ∈ R such that ∏ i |
| Databáze: | OpenAIRE |
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