On the axiomatic rank of varieties generated by a semigroup or monoid with one defining relation
A variety is said to have finite axiomatic rank if it has a system of defining identities in which the number of variables does not exceed some fixed positive integer. A variety is said to be finitely based if it can be defined by a finite number of identities. The authors' main result is that the variety generated by a finitely-generated semigroup (or monoid) with one defining relation has finite axiomatic rank if and only if it is finitely based. He also solves two related problems: he shows that the variety generated by the bicyclic monoid has infinite axiomatic rank; and he gives an example of a finitely based monoid variety generated by a monoid obtained by the adjunction of an identity element to a semigroup, but where the smaller semigroup itself generates a semigroup variety that is not finitely based.
scientific article; zbMATH DE number 4097666
scientific article