What is the Alphabetical Inequality Test?

Question

Mathematica can sort {"b","d","a","c"} into {"a","b","c","d"} with a simple application of Sort. What ordering function/command is being used?

It isn't <: "a" < "b" doesn't return True or False, just a < b.

I'm trying to sort a large data set that contains both strings and numeric values. An example:

data = {{9, 8, "b"}, {4, 2, "d"}, {0, 3, "a"}, {4, 9, "c"}}.

I can sort by the second element of each set:

Sort[data, #2[[2]] > #1[[2]] &] returns

{{4, 2, "d"}, {0, 3, "a"}, {9, 8, "b"}, {4, 9, "c"}}.

But Sorting by the third element doesn't work:

Sort[data, #2[[3]] > #1[[3]] &] returns data unchanged:

{{9, 8, "b"}, {4, 2, "d"}, {0, 3, "a"}, {4, 9, "c"}}.

It feels like a hack, but I can sort using Ordering and OrderedQ:

data[[Ordering[data[[All, 3]]]]] returns

{{0, 3, "a"}, {9, 8, "b"}, {4, 9, "c"}, {4, 2, "d"}}, as does

Sort[data, OrderedQ[{#1[[3]], #2[[3]]}] &].

Neither feels natural. Is there a lexicographical/alphabetical "inequality" command, or is it just OrderedQ?

Answer 1

You can use AlphabeticOrder:

data = {{9, 8, "b"}, {4, 2, "d"}, {0, 3, "a"}, {4, 9, "c"}};

Sort[
    data, 
    AlphabeticOrder[ #1[[3]], #2[[3]] ]&
]

{{0, 3, "a"}, {9, 8, "b"}, {4, 9, "c"}, {4, 2, "d"}}

Answer 2

AlphabeticOrder does not exist in version 10.1 which I use, but I don't believe it is necessary here. I argue that Order and OrderedQ are the more canonical (and certainly general) functions.

Additionally, where possible you should make use of SortBy (e.g. SortBy[data, #[[3]] &]) or Ordering (as you already did) rather than Sort, as these are more efficient methods.

An alternative to OrderedQ if you have need of Sort itself:

Sort[data, 0 < Order[#[[3]], #2[[3]]] &]
Sort[data, 0 > Order[#[[3]], #2[[3]]] &]

{{0, 3, "a"}, {9, 8, "b"}, {4, 9, "c"}, {4, 2, "d"}}
{{4, 2, "d"}, {4, 9, "c"}, {9, 8, "b"}, {0, 3, "a"}}

This is actually more concise than AlphabeticOrder.