# Lattice and Hopf algebra of integer relations

Abstract: We present how many well known structures such as the weak order on permutations and the Malvenuto Reutenaurer Hopf algebra can be derived from very simple definitions on the integer relations level. We define both a lattice and a Hopf alegbra on integer relations which gives us in turn both a lattice and Hopf algebra on integer posets. We see that many combinatorial objects, such as permutations and binary trees, can be represented as integer posets which leads us to re-discover some well known algebraic structures on those objects as well as new ones.