We prove for finite, graded, bounded posets, that the Poincaré-extended a b ab -index is obtained from the a b ab -index via the ω ω -transformation as conjectured by Dorpalen-Barry, Maglione, and the second author. This ω ω -transformation is a generalization of the ( c (c - 2 d ) 2d) -index, and it also generalizes the numerical canonical decomposition of the Chow ring. It provides a conceptually new approach to extended a b ab -indices and Chow polynomials, beyond R R -labeled posets, based on a well-known identity in the incidence algebra.
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Hoster et al. (2026) studied this question.
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