Abstract
Let Σ be an arbitrary signature and ϒ be a non-empty set of operation symbols within it. A (partial) Σ-algebra is ϒ-total when all its operations in ϒ are total: these are the partly total algebras in the title, and they include total algebras and attributed graphs. In this paper we establish a necessary and sufficient condition on a pair of homomorphisms of ϒ-total Σ-algebras for the existence of a pushout complement of them. This solves the application problem for the double-pushout transformation of these kinds of structure.
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