  F-signature is an important numeric invariant of singularities in positive
characteristic that can be used to detect strong F-regularity. One would like
to have a variant that rather detects F-rationality, and there are two theories
that aim to fill this gap: F-rational signature of Hochster and Yao and dual
F-signature of Sannai. Unfortunately, several important properties of the
original F-signature are unknown for these invariants.
  We give a modification of the Hochster-Yao definition that agrees with
Sannai's dual F-signature and push further the united theory to achieve a
complete generalization of F-signature.
