While investigating the connection of absolute flatness of partially ordered monoids (briefly pomonoids) with their amalgamation, the author recently posed the question as to whether it was possible to formulate some ordered analogue of Isbell's zigzag theorem. Proving such a theorem is the primary aim of this article.
We introduce four extension properties (CEP, QEP, SCEP and SQEP) for ordered algebras, similar to the congruence extension property (CEP) and the strong congruence extension property of usual (unordered) algebras. All four properties turn out to have a description in terms of commutative squares or pullback diagrams. We then use these categorical descriptions to prove an ordered analogue of the well-known relation TP = AP + CEP, namely that a variety of ordered algebras has the ordered transferability property if and only if it has the ordered amalgamation property and QEP.
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