ON DISCRETE IDEMPOTENT PATHS

Luigi Santocanale (1, 2)

1 LIS - Laboratoire d'Informatique et des Systèmes (LIS) (Marseille, Toulon)
2 LIRICA - Logique, Interaction, Raisonnement et Inférence, Complexité, Algèbre

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.

Mots clés: Involutive residuated lattice; Quantales; join-continuous map; Semigroup; discrete path; idempotent

Domaines: Logique [math.LO]; Combinatoire [math.CO]; Logique en informatique [cs.LO]

Luigi Santocanale. ON DISCRETE IDEMPOTENT PATHS. Words 2019, Sep 2019, Loughborough, United Kingdom. pp.312--325, ⟨10.1007/978-3-030-28796-2_25⟩. ⟨hal-02153821⟩