On the join of varieties. (English) Zbl 0596.08005
An algebra A of finite similarity type \(\Omega\) containing neither nullary nor unary operations is said to be nilpotent iff it satisfies the identities \(u=v\) for all ”sufficiently long” words u and v. A variety is nilpotent iff it consists of nilpotent algebras. The main result is as follows: For any varieties \({\mathfrak M}\) and \({\mathfrak N}\) of type \(\Omega\) if \({\mathfrak N}\) is nilpotent then the variety \({\mathfrak M}\vee {\mathfrak N}\) is finitely based iff \({\mathfrak M}\) is finitely based.
Reviewer: A.D.Bol’bot
MSC:
08B05 | Equational logic, Mal’tsev conditions |
20M07 | Varieties and pseudovarieties of semigroups |
08B15 | Lattices of varieties |