In this talk, we study quotients of the magmatic operad, that is the free nonsymmetric operad generated by one binary generator.
We equip the set of these quotients with a lattice structure, defined in terms of operad morphisms, and provide an analog of the
Grassmann formula for the dimensions of these operads. We study a subset of this lattice, formed by operads that we call comb
associative operads. The latter are not stable for the lattice operations of magmatic quotients, however, we define new
operations making comb associative operads a lattice, isomorphic to the lattice of integers, equipped with the division relation and
gcd, lcm operations. Finally, we study the existence of a finite convergent presentation for comb associative operads using the
Knuth-Bendix completion procedure. In particular, the comb associative operad corresponding to 3 admits a finite 
convergent presentation, from which we deduce a complete description of its Hilbert series.