Idempotents and one-sided units: Lattice invariants and a semigroup of functors on the category of monoids

JOURNAL OF ALGEBRA

卷:560

Article

East, James¹ 

[1] Western Sydney Univ, Sch Comp Engn & Math, Ctr Res Math, Locked Bag 1797, Penrith, NSW 2751, Australia

关键词: Monoids;    Idempotents;    Units;    Lattices;    Categories;    Functors;    Invariants;   

DOI  :  10.1016/j.jalgebra.2020.06.017

来源: Elsevier

【 摘 要 】

For a monoid M, we denote by G(M) the group of units, E(M) the submonoid generated by the idempotents, and G(L)(M) and G(R)(M) the submonoids consisting of all left or right units. Writing M for the (monoidal) category of monoids, G, E, G(L) and G(R) are all (monoidal) functors M -> M. There are other natural functors associated to submonoids generated by combinations of idempotents and one- or two-sided units. The above functors generate a monoid with composition as its operation. We show that this monoid has size 15, and describe its algebraic structure. We also show how to associate certain lattice invariants to a monoid, and classify the lattices that arise in this fashion. A number of examples are discussed throughout, some of which are essential for the proofs of the main theoretical results. (C) 2020 Elsevier Inc. All rights reserved.

【 授权许可】

Free