Systems of curves on non-orientable surfaces

We show that the order of the cardinality of maximal complete 1-systems of loops on non-orientable surfaces is ||^2. In particular, we determine the exact cardinality of maximal complete 1-systems of loops on punctured projective planes. To prove these results, we show that the cardinality of maximal systems of arcs pairwise-intersecting at most once on a non-orientable surface is 2|| (||+1).