The Sheffer stroke (https://en.wikipedia.org/wiki/Sheffer_stroke) is functionally complete: any truth-functional connective (such as $\wedge, \vee, \rightarrow$, . . .) can be represented purely in terms of the stroke.
My question is about a version of the Sheffer stroke for a certain subclass of formulas:
Say that a (propositional) sentence $\varphi$ is essentially positive if it is satisfied by the assignment making every propositional variable "true." It is a good exercise to show that all such sentences can be expressed using only the connectives "$\wedge$" and "$\implies$"; see e.g. A formula $\phi$ is logically equivalent to a another formula which contains only propositional variables and the connectives $\wedge$ and $\to$. Moreover, any sentence expressible just using "$\wedge$" and "$\implies$" is essentially positive.
My question is: can we make do with a single connective? Specifically:
Is there a single truth-functional such that the sentences expressible using that functional as a connective are exactly the essentially positive formulas?
It's pretty easy to see that there isn't such an operation which is binary, just by enumerating the possibilities, but there isn't an obvious reason why the simplest such functional couldn't be 15-ary, say.
More generally,
Suppose I have a finite collection of truth-functional connectives $\{C_i: i\in I\}$. When is there a single truth-functional connective $C$ such that the sentences expressible using $C$ are exactly those expressible using $\{C_i: i\in I\}$?