Ordinal sum; Horizontal sum; Lattice-based sum; Lattice-ordered set; Posets

In this contribution, the well-known ordinal sum technique of posets is generalized by allowing for a lattice ordered index set instead of a linearly ordered index set, and we argue for the merits of this generalization. We will call such a proposed sum-type construction a lattice-based sum. Our new approach of lattice-based sum extends also the horizontal sum. We show that the lattice-based sum of posets is again a poset. Subsequently, we apply the results for constructing new lattices by investigating lattice-based sums when the summand posets are lattices. We show that under certain assumptions, the lattice-based sum of lattices will be a lattice.