  Boehmians are quotients of sequences which are constructed by using a set of
axioms. In particular, one of these axioms states that the set $S$ from which
the {\it denominator} sequences are formed should be a commutative semigroup
with respect to a binary operation. In this paper, we introduce a
generalization of abstract Boehmian space, called $G$-Boehmian space, in which
$S$ is not necessarily a commutative semigroup. Next, we provide an example of
a $G$-Boehmian space and we discuss an extension of the Hartley transform on
it. Finally, we compare the Hartley transform in this paper with the existing
works on Hartley transform of Boehmians.
