  Motivated by the fact that any nonzero $\Lambda$ can introduce a length scale
or a time scale into Einstein's theory,
$r_{\Lambda}=ct_{\Lambda}=\sqrt{3/|\Lambda|}$. Conversely, any cosmological
length scale or time scale can introduce a $\Lambda (t)$,
$\Lambda(t)=3/r^2_{\Lambda}(t)=3/(c^2t^2_{\Lambda}(t))$. In this letter, we
investigate the time varying $\Lambda(t)$ corresponding to the length scales,
including the Hubble horizon, the particle horizon and the future event
horizon, and the time scales, including the age of the universe and the
conformal time. It is found out that, in this scenario, the $\Lambda(t)$CDM
model can be taken as the unified origin of the holographic and agegraphic dark
energy models with interaction between the matter and the dark energy, where
the interacting term is determined by $Q=-\dot{\rho}_{\Lambda}$. We place
observational constraints on the $\Lambda(t)$CDM models originating from
different cosmological length scales and time scales with the recently compiled
"Union2 compilation" which consists of 557 Type Ia supernovae (SNIa) covering a
redshift range $0.015\leq z \leq 1.4$. In conclusion, an accelerating expansion
universe can be derived in the cases taking the Hubble horizon, the future
event horizon, the age of the universe and the conformal time as the length
scale or the time scale.
