The free \mathfrak{m} -lattice on the poset H

Let H={a ₀, a ₁, a ₂, b ₀, b ₁, b ₂} be the poset defined by a ₀<a ₂<a ₁, b ₀<b ₂<b ₁, a ₀<b ₁, and b ₀<a ₁. For an infinite regular cardinal , we describe the free -lattice on H. This continues the work of I. Rival and R. Wille who accomplished the same for =. In subsequent papers, we show how to apply this result to describe the free -lattice on a poset for a large class of posets, called slender posets.