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Our main result is the following: let   $X$   be a normal affine toric surface without torus factor. Then there exists a non-normal affine toric surface   $X^{\prime}$   with automorphism group isomorphic to the automorphism group of   $X$   if and only if   $X$   is different from the affine plane. As a tool, we first provide a classification of normalized additive group actions on a non-necessarily normal affine toric variety   $X$   of any dimension. Recall that normalized additive group actions on   $X$   are in correspondence with homogeneous locally nilpotent derivations on the algebra of regular functions of   $X$  . More generally, we provide a classification of homogeneous locally nilpotent derivations on the semigroup algebra of a commutative cancellative monoid.

On the Automorphism Group of Non-Necessarily Normal Affine Toric Varieties