  We investigate the equilibrium properties of a quasi-two-dimensional
degenerate boson-fermion mixture (DBFM) with a bosonic vortex-antivortex
superposed state (VAVSS) using a quantum-hydrodynamic model. We show that,
depending on the choice of parameters, the DBFM with a VAVSS can exhibit rich
phase structures. For repulsive boson-fermion (BF) interaction, the
Bose-Einstein condensate (BEC) may constitute a petal-shaped "core" inside the
honeycomb-like fermionic component, or a ring-shaped joint "shell" around the
onion-like fermionic cloud, or multiple segregated "islands" embedded in the
disc-shaped Fermi gas. For attractive BF interaction just below the threshold
for collapse, an almost complete mixing between the bosonic and fermionic
components is formed, where the fermionic component tends to mimic a bosonic
VAVSS. The influence of an anharmonic trap on the density distributions of the
DBFM with a bosonic VAVSS is discussed. In addition, a stability region for
different cases of DBFM (without vortex, with a bosonic vortex, and with a
bosonic VAVSS) with specific parameters is given.
