What is the non trivial example of monosemiring?












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I considered the definition of monosemiring as: a semiring $(R, +, .)$ is said to be a monosemiring if $x+y=xy$ $forall~x, yin R,$ where $(R,+)$ and $(R, .)$ are semi groups. I also know that distributive laws will be satisfied in a monosemiring but it seems that every semi group can also be considered to be monosemiring where distributive laws will be trivially satisfied. What is the example of semi group which cannot be extended to a monosemiring? Or What is the non trivial example of monosemiring? I am bit scared as many questions went unanswered. Please someone answer this. Thank you in advance










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  • $begingroup$
    No, distributive rule isn't trivially satisfied: it translates to the identities $abc=abac$ and $abc=acbc$. Basically, monosemigroups are exactly the semigroups satisfying these identities.
    $endgroup$
    – Berci
    Dec 24 '18 at 12:08
















1












$begingroup$


I considered the definition of monosemiring as: a semiring $(R, +, .)$ is said to be a monosemiring if $x+y=xy$ $forall~x, yin R,$ where $(R,+)$ and $(R, .)$ are semi groups. I also know that distributive laws will be satisfied in a monosemiring but it seems that every semi group can also be considered to be monosemiring where distributive laws will be trivially satisfied. What is the example of semi group which cannot be extended to a monosemiring? Or What is the non trivial example of monosemiring? I am bit scared as many questions went unanswered. Please someone answer this. Thank you in advance










share|cite|improve this question











$endgroup$












  • $begingroup$
    No, distributive rule isn't trivially satisfied: it translates to the identities $abc=abac$ and $abc=acbc$. Basically, monosemigroups are exactly the semigroups satisfying these identities.
    $endgroup$
    – Berci
    Dec 24 '18 at 12:08














1












1








1





$begingroup$


I considered the definition of monosemiring as: a semiring $(R, +, .)$ is said to be a monosemiring if $x+y=xy$ $forall~x, yin R,$ where $(R,+)$ and $(R, .)$ are semi groups. I also know that distributive laws will be satisfied in a monosemiring but it seems that every semi group can also be considered to be monosemiring where distributive laws will be trivially satisfied. What is the example of semi group which cannot be extended to a monosemiring? Or What is the non trivial example of monosemiring? I am bit scared as many questions went unanswered. Please someone answer this. Thank you in advance










share|cite|improve this question











$endgroup$




I considered the definition of monosemiring as: a semiring $(R, +, .)$ is said to be a monosemiring if $x+y=xy$ $forall~x, yin R,$ where $(R,+)$ and $(R, .)$ are semi groups. I also know that distributive laws will be satisfied in a monosemiring but it seems that every semi group can also be considered to be monosemiring where distributive laws will be trivially satisfied. What is the example of semi group which cannot be extended to a monosemiring? Or What is the non trivial example of monosemiring? I am bit scared as many questions went unanswered. Please someone answer this. Thank you in advance







examples-counterexamples semigroups semiring






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edited Dec 24 '18 at 11:20









Shaun

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11.1k113688










asked Dec 24 '18 at 11:18









getegete

847




847












  • $begingroup$
    No, distributive rule isn't trivially satisfied: it translates to the identities $abc=abac$ and $abc=acbc$. Basically, monosemigroups are exactly the semigroups satisfying these identities.
    $endgroup$
    – Berci
    Dec 24 '18 at 12:08


















  • $begingroup$
    No, distributive rule isn't trivially satisfied: it translates to the identities $abc=abac$ and $abc=acbc$. Basically, monosemigroups are exactly the semigroups satisfying these identities.
    $endgroup$
    – Berci
    Dec 24 '18 at 12:08
















$begingroup$
No, distributive rule isn't trivially satisfied: it translates to the identities $abc=abac$ and $abc=acbc$. Basically, monosemigroups are exactly the semigroups satisfying these identities.
$endgroup$
– Berci
Dec 24 '18 at 12:08




$begingroup$
No, distributive rule isn't trivially satisfied: it translates to the identities $abc=abac$ and $abc=acbc$. Basically, monosemigroups are exactly the semigroups satisfying these identities.
$endgroup$
– Berci
Dec 24 '18 at 12:08










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