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Abstract:
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The purpose of this thesis is to investigate chains in partial orders
(P(X), C), where 11} (X) is the set of domains of isomorphic substructures of
a relational structure X. Since each chain in a partial order can be extended
to a maximal one, it is enough to describe maximal chains in P(X). It is
proved that, if X is an ultrahomogeneous relational structure with non-trivial
isomorphic substructures, then each maximal chain in (P(X) U {0} , C) is
a complete, R-embeddable linear order with minimum non-isolated. If X
is a relational structure, a condition is given for X, which is sufficient for
(P(X) U {0} , C) to embed each complete, R-embeddable linear order with
minimum non-isolated as a maximal chain. It is also proved that if X is one
of the following relational structures: Rado graph, Henson graph, random
poset, ultrahomogeneous poset 1,13, or ultrahomogeneous poset C, 2 ; then L
is isomorphic to a maximal chain in (P(X) U {0} , C) if and only if L is
complete, R-embeddable with minimum non-isolated. If X is a countable
antichain or disjoint union of u complete graphs on v vertices with pv =
then L is isomorphic to a maximal chain in 0P(X) U {0} , c) if and only if
L is Boolean, R-embeddable with minimum non-isolated. |