Ultraproducts of continuous posets
H. Andr\'eka, Z. Gyenis, I. N\'emeti
TL;DR
This paper demonstrates that the property of complete additivity of functions is preserved in ultraproducts of posets, establishing it as an elementary property and contrasting with the non-preservation of completeness in ultraproducts.
Contribution
It proves that complete additivity of functions is preserved under ultraproducts of posets, highlighting its elementary nature and providing new insights into ultraproduct behavior.
Findings
Complete additivity is preserved in ultraproducts of posets.
Complete additivity is an elementary property.
Ultraproducts of nontrivial complete posets are not necessarily complete.
Abstract
It is known that nontrivial ultraproducts of complete partially ordered sets (posets) are almost never complete. We show that complete additivity of functions is preserved in ultraproducts of posets. Since failure of this property is clearly preserved by ultraproducts, this implies that complete additivity of functions is an elementary property.
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