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IBM Journal of Research and Development  
Volume 20, Number 2, Page 138 (1976)
Nontopical Issue
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Bases for Chain-complete Posets

by G. Markowsky, B. K. Rosen
Various authors (especially Scott, Egli, and Constable) have introduced concepts of “basis” for various classes of partially ordered sets (posets). This paper studies a basis concept directly analogous to the concept of a basis for a vector space. The new basis concept includes that of Egli and Constable as a special case, and one of their theorems is a corollary of our results. This paper also summarizes some previously reported but little known results of wide utility. For example, if every linearly ordered subset (chain) in a poset has a least upper bound (supremum), so does every directed subset.

Given posets P and Q, it is often useful to construct maps g:P → Q that are chain-continuous: supremums of nonempty chains are preserved. Chain-continuity is analogous to topological continuity and is generally much more difficult to verify than isotonicity: the preservation of the order relation. This paper introduces the concept of an extension basis: a subset B of P such that any isotone f:B → Q has a unique chain-continuous extension g:P → Q Two characterizations of the chain-complete posets that have extension bases are obtained. These results are then applied to the problem of constructing an extension basis for the poset [P → Q] of chain-continuous maps from P to Q, given extension bases for P and Q. This is not always possible, but it becomes possible when a mild (and independently motivated) restriction is imposed on either P or Q. A lattice structure is not needed.

Related Subjects: Mathematics; Programming, programs, and programming languages