A monotone, upper semicontinuous decomposition of a compact, Hausdorff continuum is admissible if the layers (tranches) of the irreducible subcontinua of
M
M
are contained in the elements of the decomposition. It is proved that the quotient space of an admissible decomposition is hereditarily arcwise connected and that every continuum
M
M
has a unique, minimal admissible decomposition
A
\mathcal {A}
. For hereditarily unicoherent continua
A
\mathcal {A}
is also the unique, minimal decomposition with respect to the property of having an arcwise connected quotient space. A second monotone, upper semicontinuous decomposition
G
\mathcal {G}
is constructed for hereditarily unicoherent continua that is the unique minimal decomposition with respect to having a semiaposyndetic quotient space. Then
G
\mathcal {G}
refines
G
\mathcal {G}
and
G
\mathcal {G}
refines the unique, minimal decomposition
L
\mathcal {L}
of FitzGerald and Swingle with respect to the property of having a locally connected quotient space (for hereditarily unicoherent continua).