Irreducible elements and uniquely generated algebras
This paper deals with unique generation of algebras in a variety \(V\). An algebra \(A\) is uniquely generated by a set \(G\) if \(G\) generates \(A\) and \(G\) is a subset of every set that generates \(A\). The following situations are investigated: (1) every algebra in \(V\) is uniquely generated; (2) every finite algebra in \(V\) is uniquely generated; (3) every free algebra in \(V\) is uniquely generated; (4) every nontrivial algebra in \(V\) is uniquely generated. In section 2, the authors provide instances of special properties that a variety has, if the free algebras in the variety are uniquely generated. Theorem 3.1 provides a classification of the subvarieties of the variety of commutative semigroups according to the four situations mentioned above. Section 4 provides partial answers to these questions for Ockham and De Morgan algebras. In particular, all finite De Morgan algebras that are uniquely generated are determined. Section 7 provides a listing of distinct varieties generated by two-element algebras, grouped according to which of the algebras in the variety are uniquely generated. Irreducible elements are used as a primary tool in the investigations. The paper is fairly self-contained modulo elementary knowledge of universal algebra.
scientific article; zbMATH DE number 1741678