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An alternating sign matrix, or ASM, is a $(0, \pm 1)$-matrix where thenonzero entries in each row and column alternate in sign. We generalize thisnotion to hypermatrices: an $n\times n\times n$ hypermatrix $A=[a_{ijk}]$ is an{\em alternating sign hypermatrix}, or ASHM, if each of its planes, obtained byfixing one of the three indices, is an ASM. Several results concerning ASHMsare shown, such as finding the maximum number of nonzeros of an $n\timesn\times n$ ASHM, and properties related to Latin squares. Moreover, weinvestigate completion problems, in which one asks if a subhypermatrix can becompleted (extended) into an ASHM. We show several theorems of this type.

Alternating Sign Matrices and Hypermatrices, and a Generalization of Latin Square