When n has three elements, the first two are used in the manner just stated. The third element of n breaks the frame into two subframes, leading and trailing. It specifies the number of trailing axes in the frames of x and y in which cells will be paired with corresponding cells, and in which, therefore, mismatched dimensions are forbidden. Any leading dimensions present in one trailing subframe and absent in the other - which must in this case be a whole frame -, are supplied by replicating the latter subframe. Any remaining (leading) axes of the frames are used to create subframes whose members are paired in all possible ways, in the manner of an outer product. Again, n can be excessive; any excess is dealt with as described in the next paragraphs.
Assume, just for convenience, that the first two elements of n are positive. Set
cyûn[0]ÄÒÒy and cxûn[1]ÄÒÒx
(cell ranks),
tyûn[2]Ä(ÒÒy)-cy and txûn[2]Ä(ÒÒx)-cx
(ranks of the trailing parts of the frames), and
lyû0Ó(ÒÒy)-cy+ty and lxû0Ó(ÒÒx)-cx+tx
(ranks of the leading parts of the frames).
That is, the cell specifications are honored to the extent possible, then the trailing subframe specification is honored to the extent possible, and the leading subframes are whatever is left. The function f is applied to pairs of cells of rank cy and cx taken from y and x respectively, using all combinations of subscripts for the first ly and lx axes and corresponding subscripts for the next tyÓtx axes (after possible replication of one trailing subframe to make these subframes match).
Whether the cells match is the responsibility of the operand function. The two leading subframes have no compatibility requirement. The two trailing subframes must match only as far as they both exist: specifically,
(-tyÄtx)Ù(-cy)ÕÒy ûý (-tyÄtx)Ù(-cx)ÕÒx.
Clearly, if ty is less than tx (and so the trailing subframe for y is replicated to match the shape of the trailing subframe for x), then ly is zero, and if tx is less than ty, then lx is zero.
Note that:
if n has only one element, y (f@n) x is equivalent to y (f@ n,n) x
which is equivalent to y (f@ n,n,9) x
and if n has exactly two elements, y (f@n) x is equivalent to y (f@ n,9) x.
When n has fewer than three elements, the trailing subframes are simply the frames.