Product structures in Floer theory for Lagrangian cobordisms

We construct a product on the Floer complex associated to a pair of Lagrangian cobordisms. More precisely, given three exact transverse Lagrangian cobordisms in the symplectization of a contact manifold, we define a map ${\mathfrak{m}}_2$ by a count of rigid pseudo-holomorphic disks with boundary on the cobordisms and having punctures asymptotic to intersection points and Reeb chords of the negative Legendrian ends of the cobordisms. More generally, to a $(d+1)$-tuple of exact transverse Lagrangian cobordisms we associate a map ${\mathfrak{m}}_d$ such that the family $({\mathfrak{m}}_d{)}_{d\ge 1}$ are $A_{\mathrm{\infty }}$-maps. Finally, we extend the Ekholm-Seidel isomorphism to an $A_{\mathrm{\infty }}$-morphism, giving in particular that it is a ring isomorphism.