Summary: A digital moving-average filter with nonnegative mask weights that sum to unity is a gray-scale morphological filter. As such, it possesses a representation as a maximum of erosions over its morphological basis. The present paper investigates the special properties of the digial moving- average morphological basis, these properties being, for the most part, combinatoric in nature. Shapes of digital structuring elements are related to the filter weights by means of shape classes. This is accomplished by defining a characteristic of a signal relative to the filter weights and then characterizing basis elements by means of the characteristic. The notion of shape is then extended to a more general class of morphological filters. The concept or a morphological quasi average is introduced and the power of such filters to approximate morphological filters is addressed. The effect of basis reduction is studied for a rather wide class of morphological filters, the aim being to quantify the degree of which is a reduced-basis filter approximates the original full-basis filter. The reduction analysis is then applied to the moving mean.