2025/06/16 by D. Craig Miller, Miller, Craig
Computer Science · Decision Sciences · #Advanced Algebra and Logic #FOS: Mathematics #Fuzzy and Soft Set Theory #Group Theory (math.GR) #semigroups and automata theory
paper · pdf · doi:10.48550/arxiv.2506.13617
openalex publication_date 2025/06/16 · openalex created_date 2025/10/13 · openalex updated_date 2026/07/28
We systematically study the minimal conditions on L-, R- and J-classes, denoted by ML, MR and MJ, as well as the related notions of left/right/two-sided stability, in semigroups and biacts. In particular, we investigate the behaviour of these conditions with respect to quotients, substructures and extensions. Among other results, the following are proved. The conditions ML, MR and MJ are preserved under quotients (for both semigroups and biacts), but this is not the case for one- and two-sided stability. For semigroups, the conditions ML, MR and one- and two-sided stability are inherited by subsemigroups of finite Green index and also by bi-ideals (and hence by one- and two-sided ideals). Moreover, a semigroup satisfies ML (or MR) if and only if both an ideal and the associated Rees quotient do, but the analogue of this result fails in both directions for the condition MJ, and in the reverse direction for one- and two-sided stability.