2021/12/21 by Luís Oliveira, Oliveira, Luís
Computer Science · #(Primary) 20M17 #(Secondary) 20M05 #20M10 #Combinatorics (math.CO) #FOS: Mathematics #Group Theory (math.GR) #Natural Language Processing Techniques #semigroups and automata theory
paper · pdf · doi:10.48550/arxiv.2112.11310
openalex publication_date 2021/12/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
A regular semigroup is weakly generated by a set X if it has no proper regular subsemigroups containing X. In this paper, we study the regular semigroups weakly generated by idempotents. We show there exists a regular semigroup FI(X) weakly generated by |X| idempotents such that all other regular semigroups weakly generated by |X| idempotents are homomorphic images of FI(X). The semigroup FI(X) is defined by a presentation ⟨ G(X),ρe∪ρs⟩ and its structure is studied. Although each of the sets G(X), ρe, and ρs is infinite for |X|≥ 2, we show that the word problem is decidable as each congruence class has a canonical form. If FIn denotes FI(X) for |X|=n, we prove also that FI2 contains copies of all FIn as subsemigroups. As a consequence, we conclude that (i) all regular semigroups weakly generated by a finite set of idempotents, which include all finitely idempotent generated regular semigroups, strongly divide FI2; and (ii) all finite semigroups divide FI2.