  In this paper, we introduce the expansion function $\delta$ on an $L$-module
$M$. We define and investigate a $\delta$-primary element in an $L$-module $M$.
Its characterizations and many of its properties are obtained.
$\delta_0$-primary and $\delta_1$-primary elements of an $L$-module $M$ are
related with 2-absorbing, 2-absorbing primary elements of an $L$-module $M$ to
obtain their special properties. The element $\delta_1(N)\in M$ is related to
$rad(N)\in M$, the radical element of $M$ to obtain its properties where $N\in
M$. We define a $\delta_L$-primary element in an $L$-module $M$ where
$\delta_L$ is an expansion function on $L$ and find relation among a
$\delta_L$-primary element of $M$ and its corresponding $\delta_L$-primary
element of $L$.
