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Let (R,m) be a commutative complete Gorenstein local ring and
let ï¿½? be a Gorenstein order, that is to say, ï¿½? is a maximal Cohen-Macaulay
R-module and HomR(ï¿½?,R) is a projective ï¿½?-module. The main theme of this
paper is to study the representation-theoretic properties of generalized lattices,
i.e. those ï¿½?-modules which are Gorenstein projective over R. It is proved that
ï¿½? has only finitely many isomorphism classes of indecomposable lattices if and
only if every generalized lattice is the direct sum of finitely generated ones. It is
also turn out that, if R is one-dimensional, then a generalized lattice M which
is not the direct sum of copies of a finite number of lattices, contains indecomposable
sublattices of arbitrarily large finite h-length, an invariant assigned to
each generalized lattice which measures Hom modulo projectives
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