Locally strong endomorphisms of paths

• Main Authors: Sr Arworn, U. Knauer, S. Leeratanavalee
• Format: Journal
• Published: 2018
• Subjects: Mathematics
• Online Access: https://www.scopus.com/inward/record.uri?partnerID=HzOxMe3b&scp=41549100031&origin=inward; http://cmuir.cmu.ac.th/jspui/handle/6653943832/60552
• Institution: Chiang Mai University

We determine the number of locally strong endomorphisms of directed and undirected paths-direction here is in the sense of a bipartite graph from one partition set to the other. This is done by the investigation of congruence classes, leading to the concept of a complete folding, which is used to characterize locally strong endomorphisms of paths. A congruence belongs to a locally strong endomorphism if and only if the number l of congruence classes divides the length of the original path and the points of the path are folded completely into the l classes, starting from 0 to l and then back to 0, then again back to l and so on. It turns out that for paths locally strong endomorphisms form a monoid if and only if the length of the path is prime or equal to 4 in the undirected case and in the directed case also if the length is 8. Finally some algebraic properties of these monoids are described. © 2007 Elsevier B.V. All rights reserved.