BUG: Comparisons and equality with pyarrow backend return pandas extension typed booleans instead of pyarrow booleans.

[rohanjain101]

Pandas version checks

• I have checked that this issue has not already been reported.

• I have confirmed this bug exists on the latest version of pandas.

• I have confirmed this bug exists on the main branch of pandas.

Reproducible Example

>>> pd.options.mode.dtype_backend = 'pyarrow'
>>> a = pd.Series([True], dtype="bool[pyarrow]")
>>> b = pd.Series([False], dtype="bool[pyarrow]")
>>> a
0    True
dtype: bool[pyarrow]
>>> b
0    False
dtype: bool[pyarrow]
>>> a == b
0    False
dtype: boolean
>>> c = a== b
>>> c.dtype
BooleanDtype
>>> c.dtype.__module__
'pandas.core.arrays.boolean'

Issue Description

The returned series is using pandas extension boolean typed series instead of arrow typed.

Expected Behavior

I expect the series returned to be using the arrow boolean data type instead of pandas extension type.

Installed Versions

pd.show_versions()
C:\ps_0310_is\lib\site-packages_distutils_hack_init_.py:33: UserWarning: Setuptools is replacing distutils.
warnings.warn("Setuptools is replacing distutils.")

INSTALLED VERSIONS

commit : 1a2e300
python : 3.8.10.final.0
python-bits : 64
OS : Windows
OS-release : 10
Version : 10.0.22621
machine : AMD64
processor : Intel64 Family 6 Model 85 Stepping 7, GenuineIntel
byteorder : little
LC_ALL : None
LANG : None
LOCALE : English_United States.1252

pandas : 2.0.0rc0
numpy : 1.24.2
pytz : 2022.7.1
dateutil : 2.8.2
setuptools : 67.2.0
pip : 22.3.1
Cython : None
pytest : None
hypothesis : None
sphinx : None
blosc : None
feather : None
xlsxwriter : None
lxml.etree : None
html5lib : None
pymysql : None
psycopg2 : None
jinja2 : None
IPython : None
pandas_datareader: None
bs4 : None
bottleneck : None
brotli : None
fastparquet : 2023.2.0
fsspec : 2023.3.0
gcsfs : None
matplotlib : None
numba : None
numexpr : None
odfpy : None
openpyxl : None
pandas_gbq : None
pyarrow : 11.0.0
pyreadstat : None
pyxlsb : None
s3fs : None
scipy : None
snappy : None
sqlalchemy : None
tables : None
tabulate : None
xarray : None
xlrd : None
zstandard : None
tzdata : None
qtpy : None
pyqt5 : None

[mroeschke]

#51987 was merged so, this will be fixed in the RC2 release