Retracts of rectangular distributive lattices and some related observations

2021/12/23 by Czédli, Gábor
#06B99 #06D99 #FOS: Mathematics #Rings and Algebras (math.RA)

paper · doi:10.48550/arxiv.2112.12498

Abstract

By a rectangular distributive lattice we mean the direct product of two non-singleton finite chains. We prove that the retracts (ordered by set inclusion and together with the empty set) of a rectangular distributive lattice G form a lattice, which we denote by Ret(G). Also, we describe and count the retracts of G. Some easy properties of retracts, retractions, and retraction kernels of (mainly distributive) lattices are observed and several examples are presented, including a 12-element modular lattice M such that Ret(M) is not a lattice.

Related

• Absolute retracts for finite distributive lattices and slim semimodular lattices
• On the representation of finite distributive lattices
• Distributive lattices determined by weighted double skeletons
• Retracts of Posets: The Chain_gap Property and the Selection Property are Independant
• Finite Dismantlable Semidistributive Lattices are Planar