Computational Complexity of Generators and Nongenerators in Algebra

We discuss the computational complexity of several problems concerning subsets of an algebraic structure that generate the structure. We show that the problem of determining whether a given subset X generates an algebra A is P-complete, while determining the size of the smallest generating set is NP-complete. We also consider several questions related to the Frattini subuniverse, Φ(A), of an algebra A. We show that the membership problem for Φ(A) is co-NP-complete, while the membership problems for Φ(Φ(A)), Φ(Φ(Φ(A))),... all lie in the class P‖(NP). In the analysis of any algebraic structure, determining those subsets that generate the structure frequently plays a key role. This is evident in linear algebra for example, where the discussion of bases and spanning sets forms a central element of the subject. The same is true in other branches of algebra such as group and lattice theory. Knowledge of the generating subsets of an algebra gives us information on its subalgebras, homomorphic images, automorphism group etc. As computer algebra systems become commonplace in the toolboxes of scientists (see for example GAP [11], and the “algebra calculator” [21]), questions of efficiency in the determination of generating subsets arise. In this paper we address this fundamental issue by providing completeness results for several variants of the basic question: does the subset X generate the algebra A? In particular, we consider the question of minimal generating sets and the existence of a generating set of a given cardinality. In addition to questions such as these, it is sometimes of interest to ask about the role an individual element can play in the generation of an algebra. Schmid [28] suggests classifying elements as generators, nongenerators and irreducibles. To explain these, we need the notion of the Frattini subuniverse of an algebra. The Frattini subgroup has played an important role in group theory since it was first considered by Giovanni Frattini in 1885 [10]. It was observed quite some time ago that most of the basic properties of the Frattini subgroup hold more generally in any algebraic structure. Numerous papers have discussed Frattini sublattices. For a sample, see [1, 2, 9, 27]. There are 2000 Mathematics Subject Classification. 68Q17, 08A30, 20D25.