Algorithm for semigroup bases I - Sampson, M. I.; Lipcsey, Z.; Akak, E.O.; Efiong, F.A.; Essien, M.A. (International Journal of Mathematical Analysis and Modelling, 2024)
The study of generating sets and bases in semigroup theory started with the experience in the topics in vector spaces, and so with comparison, theorems on independent generating sets of Semigroup that vary from those about independent generating sets in vector spaces due to some differences in their structures are given. The paper shows that there may exist a maximal independent set of elements in a semigroup which does not form a generating set. It reveals that there may exists independent set in a semigroup which cannot be expanded to a generating set, and that adding an element independent from an independent set may make the independent set dependent in a semigroup. Also, given is an important condition fulfilled by a Basis in a semigroup. The paper gives an interesting property that serves as a relevant test tool for independent generating sets in the developed algorithms for determining the basis of any given semigroup. The resulting algorithm works for determining the basis any semigroup including classes of semigroups.