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A commutative semiring $(S, +, \cdot, 0, 1)$ with unity is said to be a semifield if for all $a, b\in S$, $a+b=0$ implies that $a=0$ and $b=0$, and $a.b=0$ implies that either $a=0$ or, $b=0$.

I found the following statement in a literature:

"As in the case of fields of characteristic $0$, semifields of characteristic $0$ are also infinite".

I feel that the above statement is not true. Following is a counterexample:

It is noted here that every idempotent semiring has characteristic $0$. Now, the idempotent semiring $$(\lbrace 0, 1, 2, 3, 4, 5\rbrace, \max, \min, 0, 5)$$ is a semifield with $5$ as unity. This semiring has characteristic $0$, but the number of elements in it is finite. Hence the claim that a semifield of characteristic $0$ may have a finite number of elements. Please so correct me if I am wrong.

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    $\begingroup$ Could you define "characteristic"? The usual definition might have several natural extensions to semirings: (a) smallest $n$ such that $n1=0$ (where $nx$ $1+1+1\dots$ $n$ times); (b) smallest $n$ such that $n1=(2n)1$. Also in arxiv.org/abs/1709.06923, semifields are assumed to have $(S-\{0\},\cdot)$ a group, which is not satisfied by your example. $\endgroup$
    – YCor
    Commented Apr 20, 2020 at 18:08
  • $\begingroup$ @YCor I am considering the definition of characteristic as in the link provided. And also, the semifield i am considering here doesn't have multiplicative inverse. $\endgroup$
    – gete
    Commented Apr 21, 2020 at 3:19
  • $\begingroup$ So, your definition of characteristic (in your link, which is a MathSE question of yours) is equivalent to the smallest $n$ such that $n1=0$, or $0$ if no such $n$ exists. But would you provide a reference to the "statement you found in the literature"? Have you checked you're using the same definition? By the way in your example $5$ could be replaced by any number $\ge 1$, and with $5$ replaced by $1$ (namely $(\{0,1\},\max,\min,0,1)$ you even get a group removing $1$. $\endgroup$
    – YCor
    Commented Apr 21, 2020 at 6:52
  • $\begingroup$ @YCor this is the literature i am referring to google.com/url?sa=t&source=web&rct=j&url=http://… Also, please see page numbers: 30, 38, 53, 54. $\endgroup$
    – gete
    Commented Apr 21, 2020 at 12:39
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    $\begingroup$ Well Remark 1 p55 (referred in your post as "statement you found in the literature) is false and you gave an obvious counterexample (as I said, changing $5$ to $1$ is enough). I'm afraid the title of the linked monograph and its publisher house don't look very serious. $\endgroup$
    – YCor
    Commented Apr 21, 2020 at 12:50

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