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:


1. Are such extended semirings known and studied?
2. Can these algebraic structures be described by identities?


Any comments are welcome!



semiring with zero- and nonzero test
I don’t know about your first question; but for the second one, the answer is no — these structures can’t be axiomatised by algebraic identities.


If they could be, then any product of such structures, with the natural induced operations, would again be one. But this is not the case: if $S$, $T$ are any such structures with $0 \neq 1$ in each of them, then the resulting operation $\nu\_{S \times T}$ on their product will satisfy $\nu\_{S \times T}(0\_S,1\_T) = (0\_S,1\_T)$, which is equal to neither $0\_{S \times T}$ or $1\_{S \times T}$. So $\nu\_{S \times T}$ does not satisfy the desired defining property.


The big picture here is [Birkhoff’s HSP theorem](http://en.wikipedia.org/wiki/Birkhoff%27s_HSP_theorem): a class of algebraic structures, over a fixed language, can be axiomatised by algebraic identities if and only if it is closed under arbitrary products and subobjects (in categorical language: under all limits), and under direct images along homorphisms.


