On maximal antichain lattices of finite posets

This paper is devoted to maximal antichain lattices of posets of arbitrary length. Maximal antichain lattices of finite posets of length 1 have been well studied and are applied, for example, in formal concept analysis. However, there are many general properties inherent in finite posets of any length. For an arbitrary element $x$ of some poset, we introduce the notions of smallest and largest maximal antichains containing $x$, which are denoted by $m_x$ and $M_x$, respectively. We prove that the equality $A=\bigvee_{x\in A}m_x=\bigwedge_{x\in A}M_x$ holds for any maximal antichain $A$. This equality allows us to describe all irreducible elements of maximal antichain lattices. The main result of this paper is a description of all finite posets whose maximal antichain lattice is isomorphic to a given lattice. Irreducible elements play a key role in this description.