The Encyclopedia of Mathematics states that a semiring is
"A non-empty set S with two associative binary operations + and $\cdot$, satisfying the distributive laws"...
https://encyclopediaofmath.org/wiki/Semi-ring
Thus, we may expect the following four elements in a semiring:
- Additive zero $a$: $a + x = x + a=a$ for all $x$;
- Additive identity $b$: $b + x = x + b = x$ for all $x$;
- Multiplicative zero $c$: $c\cdot x = x \cdot c = c$ for all $x$;
- Multiplicative identity $d$: $d\cdot x = x \cdot d = x$ for all $x$.
But the Encyclopedia of Mathematics says:
"An additive zero in a semiring $S$ is an element $a$ such that $a + x = x + a = x$ for all $x$".
What is the name of the "true" additive zero of a semiring in this case?
Is it possible to add it into any semiring?
For example:
- $a + x = x + a = a$,
- $a \cdot x = x \cdot a = 0$ (the multiplicative zero)
for any $x$.
Is there an example of a semiring in which $0 \ne a \cdot x \ne a$ for some $x$,
where $a$ is the "true" additive zero: $a + x = x + a = a$?
Note that the definition of the notion of semiring is not yet settled. In a semiring $(S, +, \cdot)$, some authors consider that $(S, +)$ and $(S, \cdot)$ are semigroups, while others consider $(S, +)$ as a commutative monoid with identity $0$ (say). When $(S, \cdot)$ is a monoid with unity $1$ (say), the structure $(S, +, \cdot)$ is also called hemiring.
Edit: Here, $1$ is additive zero if and only if $x+1=1=1+x$ for all $x\in S$, and $0$ is multiplicative zero as $0\cdot x=0=x\cdot 0$. To your last question, the answer is yes.
For example in case of distributive lattice $(L, \vee, \wedge, 0, 1)$, where $0$ and $1$ are bottom and top elements, respectively of $L$, $1$ is additive zero. While if we take $S=\Bbb N_{0}$, the set of non- negative integers endowed with the operations $(+)$ as usual addition and $(\cdot)$ as usual multiplication on $S$, then $1$ will not be additive zero (though it is multiplicative unity).