On discrete idempotent paths
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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.
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