  The set of discrete lattice paths from (0, 0) to (n, n) with North and East
steps (i.e. words w $\in$ { 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.
