×

zbMATH — the first resource for mathematics

Convex automorphisms of a lattice. (English) Zbl 0797.06004
Let \(L\) be a lattice. A bijection \(f\) of \(L\) onto itself is said to be a convex automorphism of \(L\) if, whenever \(A\subseteq L\), then \(A\) is a convex sublattice of \(L\) if and only if so is \(f(A)\). It is proved that if \(L= L_ 1\times L_ 2\times\cdots\times L_ n\), where all \(L_ i\) are directly indecomposable, then convex automorphisms of \(L\) are just the mappings \(f\) such that \(f(x)_{\pi(i)}= f_ i(x)\), where (i) \(\pi\) is a permutation of the set \(\bar n= \{1,2,\dots,n\}\), and (ii) for each \(i\in \bar n\), \(f_ i\) is either an isomorphism or a dual isomorphism of \(L_ i\) onto \(L_{\pi(i)}\).

MSC:
06B05 Structure theory of lattices
PDF BibTeX XML Cite
Full Text: EuDML
References:
[1] KOLIBIAR M.: Intervals, convex sublattices and subdirect representations of lattices. Universal algebra and applications. Banach center publ. Vol. 9, PWN, Warszaw, 1980, pp. 335-339.
[2] MARMAZEEV V. I.: The lattice of convex sublattices of a lattice. (Russian) In: Ordered sets and lattices. Vol. 9, Saratov. Gos. Univ., Saratov, 1986, pp. 50-58. · Zbl 0711.06005
[3] MARMAZEEV V. I.: A group of automorphisms of the lattice of convex sublattices of a lattice. (Russian, English summary), Vestsï Akad. Navuk BSSR Ser. Fïz.-Mat. Navuk 6 (1988), 110-112. · Zbl 0665.06005
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.