
TL;DR
This paper demonstrates that the class of $( heta,C)$-representable posets cannot be characterized by first-order axioms, extending the result to various $(eta,eta)$-representable classes, highlighting limitations in their logical definability.
Contribution
The paper proves the non-axiomatizability of $( heta,C)$-representable posets in first-order logic and generalizes this to other $(eta,eta)$-representable classes.
Findings
The class of $( heta,C)$-representable posets is not first-order axiomatizable.
Generalization to $(eta,eta)$-representable posets for certain $eta$.
Highlights limitations of first-order logic in characterizing these classes.
Abstract
A poset is -representable if it can be embedded into a field of sets in such a way that all existing joins, and all existing \emph{finite} meets are preserved. We show that the class of -representable posets cannot be axiomatized in first order logic using the standard language of posets. We generalize this result to -representable posets for certain values of and .
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