Geodesic
On maximal antichain lattices of finite posets
Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 23 (2017) no. 3, pp. 95-104 Cet article a éte moissonné depuis la source Math-Net.Ru

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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.
Keywords: poset
Mots-clés : maximal antichain, maximal antichain lattice. 
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I. A. Derendiaev. On maximal antichain lattices of finite posets. Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 23 (2017) no. 3, pp. 95-104. http://geodesic.mathdoc.fr/item/TIMM_2017_23_3_a7/

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