Odd and even Fibonacci lattices arising from a Garside monoid

TL;DR

This paper explores two Fibonacci-based lattices derived from a Garside monoid, providing combinatorial proofs of their lattice properties and explicit formulas for their structure, linking Fibonacci numbers, Schr"oder trees, and algebraic properties.

Contribution

It introduces two Fibonacci-indexed lattices from a Garside monoid, offering combinatorial proofs and explicit formulas for their meet and join operations.

Findings

The even Fibonacci lattice is the lattice of simple elements in a Garside monoid.

The odd Fibonacci lattice is an order ideal within the even Fibonacci lattice.

The number of words for the Garside element equals a Schr"oder number.

Abstract

We study two families of lattices whose number of elements are given by the numbers in even (respectively odd) positions in the Fibonacci sequence. The even Fibonacci lattice arises as the lattice of simple elements of a Garside monoid partially ordered by left-divisibility, and the odd Fibonacci lattice is an order ideal in the even one. We give a combinatorial proof of the lattice property, relying on a description of words for the Garside element in terms of Schr\"oder trees, and on a recursive description of the even Fibonacci lattice. This yields an explicit formula to calculate meets and joins in the lattice. As a byproduct we also obtain that the number of words for the Garside element is given by a little Schr\"oder number.