Is every partially ordered semiring an idempotent?
I preferred to call here a semiring $(R, +, .)$ to be an idempotent if $x+x=x$ and $x.x=x~forall~xin R$. It is apparent that we can define certain partial order relations on an idempotent semiring. Can we define a partial order relation on a none idempotent semiring? Or, is every partially ordered semiring an idempotent ?
semiring
asked Nov 27 '18 at 15:54
gete
747
add a comment |
I preferred to call here a semiring $(R, +, .)$ to be an idempotent if $x+x=x$ and $x.x=x~forall~xin R$. It is apparent that we can define certain partial order relations on an idempotent semiring. Can we define a partial order relation on a none idempotent semiring? Or, is every partially ordered semiring an idempotent ?
semiring
asked Nov 27 '18 at 15:54
gete
747
add a comment |
I preferred to call here a semiring $(R, +, .)$ to be an idempotent if $x+x=x$ and $x.x=x~forall~xin R$. It is apparent that we can define certain partial order relations on an idempotent semiring. Can we define a partial order relation on a none idempotent semiring? Or, is every partially ordered semiring an idempotent ?
semiring
asked Nov 27 '18 at 15:54
gete
747
I preferred to call here a semiring $(R, +, .)$ to be an idempotent if $x+x=x$ and $x.x=x~forall~xin R$. It is apparent that we can define certain partial order relations on an idempotent semiring. Can we define a partial order relation on a none idempotent semiring? Or, is every partially ordered semiring an idempotent ?
semiring
semiring
asked Nov 27 '18 at 15:54
gete
747
asked Nov 27 '18 at 15:54
gete
747
asked Nov 27 '18 at 15:54
gete
747
asked Nov 27 '18 at 15:54
gete
747
asked Nov 27 '18 at 15:54
gete
747
747
add a comment |
add a comment |
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$mathbb R$ is a totally ordered semiring that isn't idempotent, for example.
I guess what made you forget about this is that the natural order on idempotent semirings is defined by $aleq b$ when $aoplus b=b$, which obviously doesn't occur in the example I gave.
If the semiring isn't additively idempotent, then you do not have reflexivity of this candidate for order. So in this sense, yes, for that relation to be reflexive, you would need the semiring to be idempotent.
answered Nov 27 '18 at 16:09
rschwieb
105k1299244
can the natural order on idempotent semiring defined by $aleq b$ when $aoplus b=b$ be equivalently written as $aoplus x=b$ for some $xin Bbb R$? Also, can the multiplicative monotonicity be written as $ax=b $ for some $xin~Bbb R$?
– gete
Nov 27 '18 at 16:33
@gete I would think it would be impossible to prove anti-symmetry, then.
– rschwieb
Nov 27 '18 at 16:52
You are right, as i am also stucked in anti-symmetricity but i found such additive monotonicity in a book"Algebraic Theory and application in computer science " by U. Hebisch and H.J Weinert, Vol-5 , chapter 5 under the heading-partially ordered semirings (page-144).
– gete
Nov 27 '18 at 17:02
@gete For example, using your proposed definition and $mathbb Z$, all elements are related to one another.
– rschwieb
Nov 27 '18 at 17:21
Sorry! would you please clearify the last comment again?
– gete
Nov 27 '18 at 17:26
|
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1 Answer
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$mathbb R$ is a totally ordered semiring that isn't idempotent, for example.
I guess what made you forget about this is that the natural order on idempotent semirings is defined by $aleq b$ when $aoplus b=b$, which obviously doesn't occur in the example I gave.
If the semiring isn't additively idempotent, then you do not have reflexivity of this candidate for order. So in this sense, yes, for that relation to be reflexive, you would need the semiring to be idempotent.
answered Nov 27 '18 at 16:09
rschwieb
105k1299244
can the natural order on idempotent semiring defined by $aleq b$ when $aoplus b=b$ be equivalently written as $aoplus x=b$ for some $xin Bbb R$? Also, can the multiplicative monotonicity be written as $ax=b $ for some $xin~Bbb R$?
– gete
Nov 27 '18 at 16:33
@gete I would think it would be impossible to prove anti-symmetry, then.
– rschwieb
Nov 27 '18 at 16:52
You are right, as i am also stucked in anti-symmetricity but i found such additive monotonicity in a book"Algebraic Theory and application in computer science " by U. Hebisch and H.J Weinert, Vol-5 , chapter 5 under the heading-partially ordered semirings (page-144).
– gete
Nov 27 '18 at 17:02
@gete For example, using your proposed definition and $mathbb Z$, all elements are related to one another.
– rschwieb
Nov 27 '18 at 17:21
Sorry! would you please clearify the last comment again?
– gete
Nov 27 '18 at 17:26
|
show 2 more comments
$mathbb R$ is a totally ordered semiring that isn't idempotent, for example.
I guess what made you forget about this is that the natural order on idempotent semirings is defined by $aleq b$ when $aoplus b=b$, which obviously doesn't occur in the example I gave.
If the semiring isn't additively idempotent, then you do not have reflexivity of this candidate for order. So in this sense, yes, for that relation to be reflexive, you would need the semiring to be idempotent.
answered Nov 27 '18 at 16:09
rschwieb
105k1299244
can the natural order on idempotent semiring defined by $aleq b$ when $aoplus b=b$ be equivalently written as $aoplus x=b$ for some $xin Bbb R$? Also, can the multiplicative monotonicity be written as $ax=b $ for some $xin~Bbb R$?
– gete
Nov 27 '18 at 16:33
@gete I would think it would be impossible to prove anti-symmetry, then.
– rschwieb
Nov 27 '18 at 16:52
You are right, as i am also stucked in anti-symmetricity but i found such additive monotonicity in a book"Algebraic Theory and application in computer science " by U. Hebisch and H.J Weinert, Vol-5 , chapter 5 under the heading-partially ordered semirings (page-144).
– gete
Nov 27 '18 at 17:02
@gete For example, using your proposed definition and $mathbb Z$, all elements are related to one another.
– rschwieb
Nov 27 '18 at 17:21
Sorry! would you please clearify the last comment again?
– gete
Nov 27 '18 at 17:26
|
show 2 more comments
$mathbb R$ is a totally ordered semiring that isn't idempotent, for example.
I guess what made you forget about this is that the natural order on idempotent semirings is defined by $aleq b$ when $aoplus b=b$, which obviously doesn't occur in the example I gave.
If the semiring isn't additively idempotent, then you do not have reflexivity of this candidate for order. So in this sense, yes, for that relation to be reflexive, you would need the semiring to be idempotent.
answered Nov 27 '18 at 16:09
rschwieb
105k1299244
$mathbb R$ is a totally ordered semiring that isn't idempotent, for example.
I guess what made you forget about this is that the natural order on idempotent semirings is defined by $aleq b$ when $aoplus b=b$, which obviously doesn't occur in the example I gave.
If the semiring isn't additively idempotent, then you do not have reflexivity of this candidate for order. So in this sense, yes, for that relation to be reflexive, you would need the semiring to be idempotent.
answered Nov 27 '18 at 16:09
rschwieb
105k1299244
answered Nov 27 '18 at 16:09
rschwieb
105k1299244
answered Nov 27 '18 at 16:09
rschwieb
105k1299244
answered Nov 27 '18 at 16:09
rschwieb
105k1299244
105k1299244
can the natural order on idempotent semiring defined by $aleq b$ when $aoplus b=b$ be equivalently written as $aoplus x=b$ for some $xin Bbb R$? Also, can the multiplicative monotonicity be written as $ax=b $ for some $xin~Bbb R$?
– gete
Nov 27 '18 at 16:33
@gete I would think it would be impossible to prove anti-symmetry, then.
– rschwieb
Nov 27 '18 at 16:52
You are right, as i am also stucked in anti-symmetricity but i found such additive monotonicity in a book"Algebraic Theory and application in computer science " by U. Hebisch and H.J Weinert, Vol-5 , chapter 5 under the heading-partially ordered semirings (page-144).
– gete
Nov 27 '18 at 17:02
@gete For example, using your proposed definition and $mathbb Z$, all elements are related to one another.
– rschwieb
Nov 27 '18 at 17:21
Sorry! would you please clearify the last comment again?
– gete
Nov 27 '18 at 17:26
|
show 2 more comments
can the natural order on idempotent semiring defined by $aleq b$ when $aoplus b=b$ be equivalently written as $aoplus x=b$ for some $xin Bbb R$? Also, can the multiplicative monotonicity be written as $ax=b $ for some $xin~Bbb R$?
– gete
Nov 27 '18 at 16:33
@gete I would think it would be impossible to prove anti-symmetry, then.
– rschwieb
Nov 27 '18 at 16:52
You are right, as i am also stucked in anti-symmetricity but i found such additive monotonicity in a book"Algebraic Theory and application in computer science " by U. Hebisch and H.J Weinert, Vol-5 , chapter 5 under the heading-partially ordered semirings (page-144).
– gete
Nov 27 '18 at 17:02
@gete For example, using your proposed definition and $mathbb Z$, all elements are related to one another.
– rschwieb
Nov 27 '18 at 17:21
Sorry! would you please clearify the last comment again?
– gete
Nov 27 '18 at 17:26
can the natural order on idempotent semiring defined by $aleq b$ when $aoplus b=b$ be equivalently written as $aoplus x=b$ for some $xin Bbb R$? Also, can the multiplicative monotonicity be written as $ax=b $ for some $xin~Bbb R$?
– gete
Nov 27 '18 at 16:33
can the natural order on idempotent semiring defined by $aleq b$ when $aoplus b=b$ be equivalently written as $aoplus x=b$ for some $xin Bbb R$? Also, can the multiplicative monotonicity be written as $ax=b $ for some $xin~Bbb R$?
– gete
Nov 27 '18 at 16:33
@gete I would think it would be impossible to prove anti-symmetry, then.
– rschwieb
Nov 27 '18 at 16:52
@gete I would think it would be impossible to prove anti-symmetry, then.
– rschwieb
Nov 27 '18 at 16:52
You are right, as i am also stucked in anti-symmetricity but i found such additive monotonicity in a book"Algebraic Theory and application in computer science " by U. Hebisch and H.J Weinert, Vol-5 , chapter 5 under the heading-partially ordered semirings (page-144).
– gete
Nov 27 '18 at 17:02
You are right, as i am also stucked in anti-symmetricity but i found such additive monotonicity in a book"Algebraic Theory and application in computer science " by U. Hebisch and H.J Weinert, Vol-5 , chapter 5 under the heading-partially ordered semirings (page-144).
– gete
Nov 27 '18 at 17:02
@gete For example, using your proposed definition and $mathbb Z$, all elements are related to one another.
– rschwieb
Nov 27 '18 at 17:21
@gete For example, using your proposed definition and $mathbb Z$, all elements are related to one another.
– rschwieb
Nov 27 '18 at 17:21
Sorry! would you please clearify the last comment again?
– gete
Nov 27 '18 at 17:26
Sorry! would you please clearify the last comment again?
– gete
Nov 27 '18 at 17:26
|
show 2 more comments
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