I.A. Derendiaev. On maximal antichain lattices of finite posets ... P. 95-104.

This paper is devoted to maximal antichain lattices of posets of arbitrary length. Maximal antichain lattices of finite posets of length1 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, maximal antichain, maximal antichain lattice.

The paper was received by the Editorial Office on May 19, 2017

Ilia Aleksandrovich Derendiaev, graduate student, Ural Federal University, Yekaterinburg, 620002 Russia,
  e-mail: ilia.derendiaev@yandex.ru 


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