OPEN ACCESS Mathematics Open Vol. 1 (2022) 2250002 (14 pages) c  The Author(s) DOI: 10.1142/S281100722250002X On automorphisms of strong semilattice of groups Aftab Hussain Shah * and Dilawar Juneed Mir † Department of Mathematics, Central University of Kashmir Ganderbal, Tulmulla, Jammu and Kashmir 191201, India * aftab@cukashmir.ac.in † mirjunaid@cukashmir.ac.in Noor Mohammad Khan Department of Mathematics, Aligarh Muslim University Aligarh, Uttar Pradesh 202002, India noormohammadkhan@gmail.com Received 17 August 2022 Revised 4 November 2022 Accepted 10 November 2022 Published 8 December 2022 In this paper, we consider the automorphisms of the strong semilattice of groups and relate them to the isomorphisms and automorphisms of underlying groups. We also provide a construction for non-trivial automorphisms of semilattices. Keywords : Automorphisms; idempotents; structure homomorphisms; semilattices of groups. Mathematics Subject Classification 2020: 20D45, 20E36, 20K30, 20M18 1. Introduction Let (Ω, ≤) be a poset. Following [2] we say that (Ω, ≤) is a lower (meet) semilattice if a ∧ b ∈ Ω for all a,b ∈ Ω. Dually, one can define an upper (join) semilattice. In this paper by a semilattice, we shall always mean a lower semilattice. It is well known (see [2, Proposition 1.3.2]) that if (Ω, ≤) is a semilattice, then a binary operation may be defined on Ω as αβ = α ∧ β for all α,β ∈ Ω under which Ω becomes a commutative semigroup of idempotents and conversely if (Ω, ∧) is a commutative semigroup of idempotents, then (Ω, ≤), where α ≤ β if and only if αβ = α for all α,β ∈ Ω, is a semilattice. * Corresponding author. This is an Open Access article published by World Scientific Publishing Company. It is distributed under the terms of the Creative Commons Attribution 4.0 (CC BY) License which permits use, distribution and reproduction in any medium, provided the original work is properly cited. 2250002-1 Math. Open Downloaded from www.worldscientific.com by 54.152.192.173 on 01/07/23. Re-use and distribution is strictly not permitted, except for Open Access articles.