A property of meets in slim semimodular lattices and its application to retracts
| Autor: | Cz��dli, G��bor |
|---|---|
| Rok vydání: | 2021 |
| Předmět: | |
| DOI: | 10.48550/arxiv.2112.07594 |
| Popis: | Slim semimodular lattices were introduced by G. Gr��tzer 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. 13 pages, 8 figures |
| Databáze: | OpenAIRE |
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