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There are basic ways to qualitatively classify deviations from bijectivity of a function $f: x \to y$, e.g. non-injective, non-surjective, non-existence of an inverse: more generally non-monomorphic, non-epimorphic.

Are there two "natural" quantitative measures of deviation from injectivity and surjectivity?

Is there one natural quantitative measure of "total deviation" from bijectivity (combining both injective/surjectivity violation)? And from isomorphic?

Brain storm: Relative entropy of $\{f^{-1}(y) \}$, i.e. the preimages of $f$, seems one relevant to measuring relative injectivity? Relative measure or cardinality $|Im(f)/Cod(f)|$ seems one way to measure surjectivity? These both invoke additional concepts, e.g. probability or measure. I am sure mathematicians will have clearer and better ways that I can't think of. Any ideas or references will be most welcome.

The above mostly relates to functions between sets. How does one ask and answer the corresponding questions for structure-preserving functions, i.e. functorial functions, such as monotone, equivariant, homomorphic, continuous? (Apologies if this latter is asking too much in one question!).

Thanks!

JRC
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  • A possible track: Jaccard's index of likelyhood and related matrices as described in a question of mine. – Jean Marie Nov 02 '20 at 10:43
  • @JRC Given an arbitrary map $f \colon A \to B$, the family of cardinals $\left(\left|f^{－1}[{y}]\right|\right)_{y \in B}$ encodes very precisely the bijectivity of $f$. One null component in this family equivalates to lack of surjectivity and similarly one component at least $2$ equivalates to lack of injectivity. – ΑΘΩ Nov 02 '20 at 13:06
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    @ΑΘΩ Thanks. However this is still quite qualitative. I am looking for a single summary, e.g. a number that quantifies the extent of violation. – JRC Nov 02 '20 at 14:58
  • @JRC To which I would say that's too much to expect in general. – ΑΘΩ Nov 02 '20 at 15:37
  • @ΑΘΩ For my understanding. I actually didn't know that the "preimage of values outside of the image", as used in your family of cardinals, was even defined. (So how can we examine it's cardinality?) Perhaps it is convention to send values outside the image to the empty set of the domain? This empty set has cardinality 0. Under this condition, the sum of absolute deviations from 1 (over all preimage members) gives one integer measure of non-bijectivity. – JRC Nov 03 '20 at 12:55
  • @JRC From a very general point of view, since maps are constructed partly from ingredients called graphics - which have sections and inverse images over any set - it wouldn't be conceptually impossible to speak about preimages of a map over an arbitrary set. The custom and the tradition is however to consider preimages through a map $f \colon A \to B$ of any subset $Y \subseteq B$ of the codomain. $Y$ does not have to be limited to the image $\mathrm{Im}f$. – ΑΘΩ Nov 03 '20 at 14:07
  • @JRC I must also add that whenever $y \in B \setminus \mathrm{Im}f$ it is not by convention that we say $f^{-1}[{y}]＝\varnothing$, but this is in fact a theorem of set theory, a result perfectly provable starting from the axioms and the general definition of preimages. – ΑΘΩ Nov 03 '20 at 14:10

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Not a metric but, the following simply allows you to partially order functions by injectivity.

There is a partial order on the inverse image equivalence relations induced by the set of all functions $\{f|f:A \to B\}$, i.e. where each class of the relation or quotient on domain $A$ induced by $f$ is defined by $ x\equiv_fy$ iff $f(x)=f(y)$, for $x,y \in A$. The partial order encodes how coarse or fine the equivalence relations are, relative to one another, which in turn encodes nothing but the relative injectivity. A limitation is that this is not a total order: two "equally injective" functions may be incomparable and you would never know.

In terms of a numeric measure, for functions between finite sets, I guess you can also just calculate the entropy on the family of cardinals $(|f^{-1}(y)|)_{y \in B}$, treating the latter as a probability distribution (normalized frequency histogram). Not sure currently how this generalizes.

JRC
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