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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 \(L_{\pi(i)}\).

06B05 Structure theory of lattices
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[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
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