Let $\mathcal{S}=(S,\oplus,\otimes,0,1)$ be a commutative semiring and define functions $\nu:S\to \lbrace 0,1\rbrace$ and $\bar\nu:S\to \lbrace 0,1\rbrace$ as: $$ \text{$\nu(s)=0$ if $s=0$; and $\nu(s)=1$ otherwise} $$ and $$ \text{$\bar\nu(s)=1$ if $s=0$; and $\bar\nu(s)=0$ otherwise}. $$ Consider $\mathcal{S}$ extended with $\nu$ and $\bar\nu$, that is, $(S,\oplus,\otimes,\nu,\bar\nu,0,1)$.

I have the following questions:

Are such extended semirings known and studied?

Can these algebraic structures be described by identities?

Any comments are welcome!