Finitely determined members of varieties of groups and rings
Küçük Resim Yok
Tarih
2000
Yazarlar
Dergi Başlığı
Dergi ISSN
Cilt Başlığı
Yayıncı
Academic Press Inc
Erişim Hakkı
info:eu-repo/semantics/openAccess
Özet
A finitely generated algebra A in a variety V is called finitely determined in V if there exists a finite V-consistent set of equalities and inequalities in an alphabet containing the generating set of A, which, together with the identities of V, yields all relations and non-relations of A. Obviously, if the equational theory of V is recursively enumerable then any finitely determined algebra in V has solvable word problem. The known algebraic characterizations of groups and semigroups with solvable word problem imply that in the varieties of all groups and all semigroups the members with solvable word problem are finitely determined. We construct a finitely generated center-by-metabelian group with solvable word problem, which is not finitely determined in every group variety V with ZU(2) subset of or equal to V subset of or equal to U-3. We show that every extension of a finitely generated abelian group by a finite group from a variety W is finitely determined in every variety V superset of or equal to ZU(2)W. However, in any abelian-by-nilpotent variety no infinite group is finitely determined; moreover, in every variety, in which all finitely presented algebras are residually finite, each finitely determined algebra is finite. In the variety of all associative linear algebras over a finitely generated field every member with solvable word problem is finitely determined. We construct an example, which shows that for the variety of all associative rings it is not true; however, in this variety each torsion-free member with solvable word problem is finitely determined. (C) 2000 Academic Press.
Açıklama
Anahtar Kelimeler
Variety Of Algebras, Finitely Determined Algebra, Existentially Closed Algebra, Word Problem, Center-By-Metabelian Group, Associative Ring
Kaynak
Journal of Algebra
WoS Q Değeri
Q2
Scopus Q Değeri
Q1
Cilt
228
Sayı
2
