A proper merging of two disjoint quasi-ordered sets P and Q is a
quasi-order on the union of P and Q such that the restriction to P or Q
yields the original quasi-order again and such that no elements of P and Q
are identified. In this article, we determine the number of proper mergings in
the case where P is a star (i.e. an antichain with a smallest element
adjoined), and Q is a chain. We show that the lattice of proper mergings of
an m-antichain and an n-chain, previously investigated by the author, is a
quotient lattice of the lattice of proper mergings of an m-star and an
n-chain, and we determine the number of proper mergings of an m-star and an
n-chain by counting the number of congruence classes and by determining their
cardinalities. Additionally, we compute the number of Galois connections
between certain modified Boolean lattices and chains.Comment: 27 pages, 7 figures, 1 table. Jonathan Farley has solved Problem
4.18; added Section 4.4 to describe his solutio