Recursively defined data types: part 1
POPL '73 Proceedings of the 1st annual ACM SIGACT-SIGPLAN symposium on Principles of programming languages
Computability concepts for programming language semantics
STOC '75 Proceedings of seventh annual ACM symposium on Theory of computing
Correct and optimal implementations of recursion in a simple programming language
STOC '73 Proceedings of the fifth annual ACM symposium on Theory of computing
SWAT '74 Proceedings of the 15th Annual Symposium on Switching and Automata Theory (swat 1974)
Mechanizable proofs about parallel processes
SWAT '73 Proceedings of the 14th Annual Symposium on Switching and Automata Theory (swat 1973)
Program equivalence and context-free grammars
Journal of Computer and System Sciences
Hi-index | 0.00 |
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 isofonicity: 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.