Abstract
A structure is called homogeneous if every isomorphism between finite substructures of the structure extends to an automorphism of the structure. Recently, P. J. Cameron and J. Nešetřil introduced a relaxed version of homogeneity: we say that a structure is homomorphism-homogeneous if every homomorphism between finite substructures of the structure extends to an endomorphism of the structure. In this paper we characterize homomorphism-homogeneous partially ordered sets (where a homomorphism between partially ordered sets A and B is a mapping f : A →B satisfying \(x \leqslant y \Rightarrow f{\left( x \right)} \leqslant f{\left( y \right)}\)). We show that there are five types of homomorphism-homogeneous partially ordered sets: partially ordered sets whose connected components are chains; trees; dual trees; partially ordered sets which split into a tree and a dual tree; and X 5-dense locally bounded partially ordered sets.
Similar content being viewed by others
References
Cameron, P.J., Nešetřil, J.: Homomorphism-homogeneous relational structures. Comb. Probab. Comput. 15, 91–103 (2006)
Grabowski, J.-U.: Varieties and Clones of Relational Structures. PhD Thesis, Dresden University of Technology (2002)
Lockett, D.: Symmetry in the Infinite: Posets, homomorphisms, and homogeneity. Ann Cook Prize entry, October (2006). http://www.maths.qmul.ac.uk/postgraduate/anncook/lockett06.pdf
Schmerl, J.H.: Countable homogeneous partially ordered sets. Algebra Univers. 9, 317–321 (1979)
Author information
Authors and Affiliations
Corresponding author
Additional information
Supported by the Ministry od Science and Environmental Protection of the Republic of Serbia, Grant No. 144017.
Rights and permissions
About this article
Cite this article
Mašulović, D. Homomorphism-Homogeneous Partially Ordered Sets. Order 24, 215–226 (2007). https://doi.org/10.1007/s11083-007-9069-x
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11083-007-9069-x