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SUM-ORDERED PARTIAL SEMIRINGS
If we endow the set of partial functions from a data set to itself with an addition (disjoint-domain sums) and a multiplication (functional composition), then any iterative algorithm may be described formally as the solution to a matrix equation, where the matrix entries are partial functions which describe the parts of the algorithm. This suggests that algorithms may be transformed by manipulation of matrices of partial functions. Hence, it becomes necessary to understand how such matrices behave. The partial functions under disjoint-domain sums and functional composition do not form a field, and thus conventional linear algebra is not applicable. However, they can be regarded as a sum-ordered partial semiring or "so-ring", an algebraic structure possessing a natural partial ordering, an infinitary partial addition, and a binary multiplication, subject to a set of axioms. The majority of this dissertation is devoted to a detailed study of the properties and interesting substructures of so-rings themselves; preliminary results illustrating the behavior of matrices over so-rings are also presented. We hope that this study in part provides a basis for a matrix theory of algorithm transformation. ^
STEENSTRUP, MARTHA EDMAY, "SUM-ORDERED PARTIAL SEMIRINGS" (1985). Doctoral Dissertations Available from Proquest. AAI8509606.