Characters on inverse semigroups: Hahn-Banach?

Question

Let $S$ be an inverse semigroup with zero $0\in S$, $$ 0s=0=s0,\quad\forall s\in S. $$ Let $e=e^2\in S$ be an idempotent and consider the character, $$ \phi:U\to\{0,1\}:\quad\phi(e)=1,\quad\phi(0)=0, $$ defined on the subsemigroup $U:=\{e,0\}\leq S$.
Is it possible to extend the character to the whole inverse semigroup, $\phi:S\to\{0,1\}$?

Answer 1

No. Take the "universal counterexample", aka the inverse semigroup $\{a, b, ab, ba, 0\}$ defined by the relations $aba = a$, $bab = b$ and $aa = bb = 0$. Let $e = ab$. Since $1 = \varphi(e) = \varphi(ab) = \varphi(a)\varphi(b)$, one gets $\varphi(a) = \varphi(b) = 1$, whence $\varphi(aa) = \varphi(a)^2 = 1$. But $aa = 0$ and hence $\varphi(aa) = 0$. Contradiction.