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

##### MSC:
 06B05 Structure theory of lattices
##### Keywords:
convex isomorphism; convex automorphism; convex sublattice
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##### References:
  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.  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  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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