Irreducible elements and uniquely generated algebras

An algebra A is uniquely generated by a set G if G is a subset of every set that generates A. We investigate uniquely generated algebras and focus especially on equational classes of algebras in which the free algebras are uniquely generated. We show that such classes possess a number of algebraic properties that are in some sense extremal. We also present algebraic conditions on an equational class that force free algebras in the class to be uniquely generated. Various familiar equational classes are analyzed with respect to the occurrence of uniquely generated algebras in them; the final section classifies all equational classes generated by two-element algebras from this point of view. Throughout we use irreducible elements as a tool in our investigations. (C) 2002 Elsevier Science B.V. All rights reserved.