  We study the regularity and the algebraic properties of certain lattice
ideals. We establish a map I --> I\~ between the family of graded lattice
ideals in an N-graded polynomial ring over a field K and the family of graded
lattice ideals in a polynomial ring with the standard grading. This map is
shown to preserve the complete intersection property and the regularity of I
but not the degree. We relate the Hilbert series and the generators of I and
I\~. If dim(I)=1, we relate the degrees of I and I\~. It is shown that the
regularity of certain lattice ideals is additive in a certain sense. Then, we
give some applications. For finite fields, we give a formula for the regularity
of the vanishing ideal of a degenerate torus in terms of the Frobenius number
of a semigroup. We construct vanishing ideals, over finite fields, with
prescribed regularity and degree of a certain type. Let X be a subset of a
projective space over a field K. It is shown that the vanishing ideal of X is a
lattice ideal of dimension 1 if and only if X is a finite subgroup of a
projective torus. For finite fields, it is shown that X is a subgroup of a
projective torus if and only if X is parameterized by monomials. We express the
regularity of the vanishing ideal over a bipartie graph in terms of the
regularities of the vanishing ideals of the blocks of the graph.
