A property of meets in slim semimodular lattices and its application to retracts

TL;DR

This paper investigates the structure of meets in slim semimodular lattices, proving a property about intervals formed by incomparable elements, and explores how this property influences the characterization of sublattice retracts.

Contribution

It establishes a new property of meets in slim semimodular lattices and applies this to describe absorption properties of their retracts.

Findings

Intervals between incomparable elements form chains of normal slopes.

Most elements in these chains are meet-reducible.

Sublattice retracts satisfy specific absorption properties.

Abstract

Slim semimodular lattices were introduced by G. Gr\"atzer and E. Knapp in 2007, and they have intensively been studied since then. It is often reasonable to give these lattices by their $\mathcal C_1$-diagrams defined by the author in 2017. We prove that if $x$ and $y$ are incomparable elements in such a lattice $L$, then the interval $[x\wedge y, x]$ is a chain and this chain is of a normal slope in every $\mathcal C_1$-diagram of $L$. Except possibly for $x$, the elements of this chain are meet-reducible. If $A $ and $X$ are subsets of a lattice $K$, then a sublattice $S$ of a lattice $L$ has the absorption property $(K,A ,X)$ if for every embedding $g: K\to L$ such that $g(A)\subseteq S$, we have that $g(X)\subseteq S$. If there is an idempotent endomorphism $f: L\to L$ such that $S=f(L)$, then the sublattice $S$ is a retract of $L$. Applying the above-mentioned property of meets, we present two absorption properties that the retracts of every slim semimodular lattice $L$ have.