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

Abstract

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 $\mathfrak{m}$, we describe the free $\mathfrak{m}$-lattice on H. This continues the work of I. Rival and R. Wille who accomplished the same for $\mathfrak{m}$=ℵ. In subsequent papers, we show how to apply this result to describe the free $\mathfrak{m}$-lattice on a poset for a large class of posets, called slender posets.