Prove that a semiring with identity $xy=x+y$ if and only if $x=y$ is an idempotent.
$begingroup$
A semiring $R$ with an identity $xy=x+y$ if and only if $x=y$ for all $x, yin R$ is an idempotent. The counter examples of such semirings are easy to find as for example if $Bbb Z$ is a set of positive integers and the binary operations defined on $Bbb Z$ are $ab=min(a,b)$ and $a+b=max(a,b)$. Then $(R, +, .)$ is an idempotent semiring satisfying the given identity. I want a general proof that a semiring with such identity is an idempotent.
semiring
asked Nov 24 '18 at 16:55
getegete
747
$endgroup$
add a comment |
$begingroup$
A semiring $R$ with an identity $xy=x+y$ if and only if $x=y$ for all $x, yin R$ is an idempotent. The counter examples of such semirings are easy to find as for example if $Bbb Z$ is a set of positive integers and the binary operations defined on $Bbb Z$ are $ab=min(a,b)$ and $a+b=max(a,b)$. Then $(R, +, .)$ is an idempotent semiring satisfying the given identity. I want a general proof that a semiring with such identity is an idempotent.
semiring
asked Nov 24 '18 at 16:55
getegete
747
$endgroup$
add a comment |
$begingroup$
A semiring $R$ with an identity $xy=x+y$ if and only if $x=y$ for all $x, yin R$ is an idempotent. The counter examples of such semirings are easy to find as for example if $Bbb Z$ is a set of positive integers and the binary operations defined on $Bbb Z$ are $ab=min(a,b)$ and $a+b=max(a,b)$. Then $(R, +, .)$ is an idempotent semiring satisfying the given identity. I want a general proof that a semiring with such identity is an idempotent.
semiring
asked Nov 24 '18 at 16:55
getegete
747
$endgroup$
A semiring $R$ with an identity $xy=x+y$ if and only if $x=y$ for all $x, yin R$ is an idempotent. The counter examples of such semirings are easy to find as for example if $Bbb Z$ is a set of positive integers and the binary operations defined on $Bbb Z$ are $ab=min(a,b)$ and $a+b=max(a,b)$. Then $(R, +, .)$ is an idempotent semiring satisfying the given identity. I want a general proof that a semiring with such identity is an idempotent.
semiring
semiring
asked Nov 24 '18 at 16:55
getegete
747
asked Nov 24 '18 at 16:55
getegete
747
edited Nov 24 '18 at 17:43
gete
asked Nov 24 '18 at 16:55
getegete
747
asked Nov 24 '18 at 16:55
getegete
747
asked Nov 24 '18 at 16:55
getegete
747
747
add a comment |
add a comment |
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