Sorry for my bad English.
I'm trouble about a definition of an affine toric varieties.
I often see a definition of affine toric varieties as follow;
An affine toric variety is an irreducible affine variety $V$ containing a torus $T_N$ as a Zariski open subset such that the action on $T_N$ on itself extends to algebraic action of $T_N$ on V.
Now I confuse what is data about toric variety. When we say $V$ is a toric variety, dose it include date about a lattice $N$, an open immersion $T_N\to V,$and an action $T_N$ on $V$?
Or simply is toric variety a property about an algebraic variety?