On Nilpotent Extensions of Algebras

• 来源期刊: 代数集刊（英文版）
• 关键词: normal identity normalization of an algebra nilpotent extension of an algebra pn-sequence of an algebra or a variety variety of algebras nilpotent shift of a variety,generic (minimal generic) of a variety
• 年，卷（期）: 2007,（4）
• 页码范围: 593-604
• 页数: 12页
• 语种: 英文

摘要

In this paper, we investigate essentially n-ary term operations of nilpotent extensions of algebras. We detect the connection between term operations of an original algebra and its nilpotent extensions. This structural point of view easily leads to the conclusion that the number of distinct essentially n-ary term operations of a proper algebraic nilpotent extension (ひ) of an algebra (ワ) is given by the formula pn(ひ)={pn(ワ)+1 for n=1,{pn(ワ) otherwise. We show that in general the converse theorem is not true. However, we suppose that if a variety V is uniquely determined by its pn-sequences, the converse theorem is also satisfied. In the second part of the paper, we characterize generics of nilpotent shifts of varieties and describe cardinalities of minimal generics. We give a number of examples and pose some problems.