ON DISCRETE IDEMPOTENT PATHS

Abstract : The set of discrete lattice paths from (0, 0) to (n, n) with North and East steps (i.e. words w ∈ { x, y } * such that |w| x = |w| y = n) has a canonical monoid structure inherited from the bijection with the set of join-continuous maps from the chain { 0, 1,. .. , n } to itself. We explicitly describe this monoid structure and, relying on a general characterization of idempotent join-continuous maps from a complete lattice to itself, we characterize idempotent paths as upper zigzag paths. We argue that these paths are counted by the odd Fibonacci numbers. Our method yields a geometric/combinatorial proof of counting results, due to Howie and to Laradji and Umar, for idempotents in monoids of monotone endomaps on finite chains.
Complete list of metadatas

Cited literature [21 references]  Display  Hide  Download

https://hal.archives-ouvertes.fr/hal-02153821
Contributor : Luigi Santocanale <>
Submitted on : Wednesday, June 12, 2019 - 3:12:17 PM
Last modification on : Friday, June 14, 2019 - 2:00:02 AM

Files

0.pdf
Files produced by the author(s)

Identifiers

  • HAL Id : hal-02153821, version 1
  • ARXIV : 1906.05590

Collections

Citation

Luigi Santocanale. ON DISCRETE IDEMPOTENT PATHS. Words 2019, Sep 2019, Loughborough, United Kingdom. ⟨hal-02153821⟩

Share

Metrics

Record views

39

Files downloads

10