  This paper continues the study of combinatorial properties of binary
functions --- that is, functions $f:2^E\rightarrow\mathbb{C}$ such that
$f(\emptyset)=1$, where $E$ is a finite set. Binary functions have previously
been shown to admit families of transforms that generalise duality, including a
trinity transform, and families of associated minor operations that generalise
deletion and contraction, with both these families parameterised by the complex
numbers. Binary function representations exist for graphs (via the indicator
functions of their cutset spaces) and indeed arbitrary matroids (as shown by
the author previously). In this paper, we characterise degenerate elements ---
analogues of loops and coloops --- in binary functions, with respect to any
pair of minor operations from our complex-parameterised family. We then apply
this to study the relationship between binary functions and Tutte's alternating
dimaps, which also support a trinity transform and three associated minor
operations. It is shown that only the simplest alternating dimaps have binary
representations of the form we consider, which seems to be the most direct type
of representation. The question of whether there exist other, more
sophisticated types of binary function representations for alternating dimaps
is left open.
