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Let
$V$
be a vector space and let
$T(V)$
denote the semigroup (under composition) of all linear transformations from
$V$
into
$V$
. For a fixed subspace
$W$
of
$V$
, let
$T(V,W)$
be the semigroup consisting of all linear transformations from
$V$
into
$W$
. In 2008, Sullivan [‘Semigroups of linear transformations with restricted range’, Bull. Aust. Math. Soc.
77(3) (2008), 441–453] proved that
is the largest regular subsemigroup of
$$\begin{eqnarray}\displaystyle Q=\{\unicode[STIX]{x1D6FC}\in T(V,W):V\unicode[STIX]{x1D6FC}\subseteq W\unicode[STIX]{x1D6FC}\} & & \displaystyle \nonumber\end{eqnarray}$$
$T(V,W)$
and characterized Green’s relations on
$T(V,W)$
. In this paper, we determine all the maximal regular subsemigroups of
$Q$
when
$W$
is a finite-dimensional subspace of
$V$
over a finite field. Moreover, we compute the rank and idempotent rank of
$Q$
when
$W$
is an
$n$
-dimensional subspace of an
$m$
-dimensional vector space
$V$
over a finite field
$F$
.
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