Computational Complexity of Generators and Nongenerators in Algebra

Abstract. 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, (A), of an algebra A. We show that the membership problem for (A) is co-NP-complete, while the membership problems for ((A)), (((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, homomor...

• Type: Journal
• Year: 2002
• Where: IJAC
• Authors: Clifford Bergman, Giora Slutzki
• Keywords: Algebra, Algebraic Structure, Generating Set, IJAC 2002