Different `%between%` behavior on character vectors undocumented

[ToeKneeFan]

At the moment, the help documentation for %between% states that

between is equivalent to x >= lower & x <= upper when incbounds=TRUE, or x > lower & y < upper when FALSE. With a caveat that NA in lower or upper are taken as a missing bound and return TRUE not NA.

While this appears to be true for numeric vectors, for character vectors, NA is not treated as a missing bound and yields NA, as seen below, and this difference is not noted in the documentation. Since (theoretically) the lexicographic order is a total order, one might expect the same or similar behavior unless indicated otherwise, and the documentation appears to imply this, given it does not mention character vectors are treated differently. The different behavior on character vectors should probably be either (a) documented if this is intended or (b) altered to treat NA as a missing bound (as with numeric vectors).

# Minimal reproducible example]

library(data.table)

numbers <- 1:26
numbers %between% c(13, NA)
#  [1] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE  TRUE
# [14]  TRUE  TRUE  TRUE  TRUE  TRUE  TRUE  TRUE  TRUE  TRUE  TRUE  TRUE  TRUE  TRUE
letters %between% c("m", NA_character_)
#  [1] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE    NA
# [14]    NA    NA    NA    NA    NA    NA    NA    NA    NA    NA    NA    NA    NA

# Output of sessionInfo()

R version 3.6.0 (2019-04-26)
Platform: x86_64-apple-darwin15.6.0 (64-bit)
Running under: macOS Mojave 10.14.5

Matrix products: default
BLAS:   /System/Library/Frameworks/Accelerate.framework/Versions/A/Frameworks/vecLib.framework/Versions/A/libBLAS.dylib
LAPACK: /Library/Frameworks/R.framework/Versions/3.6/Resources/lib/libRlapack.dylib

locale:
[1] en_US.UTF-8/en_US.UTF-8/en_US.UTF-8/C/en_US.UTF-8/en_US.UTF-8

attached base packages:
[1] stats     graphics  grDevices utils     datasets  methods   base     

other attached packages:
[1] data.table_1.12.3

loaded via a namespace (and not attached):
[1] compiler_3.6.0 tools_3.6.0   

data.table package is the most recent stable development version pulled today using install.packages.