Name for a field without the requirement of having an additive inverse?
$begingroup$
I find this structure to be useful in my work, but I cannot find any name for it. I think it might be a "commutative division semiring" but I'm not certain because I cannot find any literature on it.
Basically, I'm looking for a name of some set $S$ where $S$ forms a commutative monoid under addition, the nonzero elements of $S$ an abelian group under multiplication, and the distributive law holds. An example is the set ${False,True}$ with logical OR as addition and logical AND as multiplication.
abstract-algebra ring-theory field-theory terminology
asked Dec 6 '18 at 13:22
GarmekainGarmekain
1,323720
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add a comment |
$begingroup$
I find this structure to be useful in my work, but I cannot find any name for it. I think it might be a "commutative division semiring" but I'm not certain because I cannot find any literature on it.
Basically, I'm looking for a name of some set $S$ where $S$ forms a commutative monoid under addition, the nonzero elements of $S$ an abelian group under multiplication, and the distributive law holds. An example is the set ${False,True}$ with logical OR as addition and logical AND as multiplication.
abstract-algebra ring-theory field-theory terminology
asked Dec 6 '18 at 13:22
GarmekainGarmekain
1,323720
$endgroup$
add a comment |
$begingroup$
I find this structure to be useful in my work, but I cannot find any name for it. I think it might be a "commutative division semiring" but I'm not certain because I cannot find any literature on it.
Basically, I'm looking for a name of some set $S$ where $S$ forms a commutative monoid under addition, the nonzero elements of $S$ an abelian group under multiplication, and the distributive law holds. An example is the set ${False,True}$ with logical OR as addition and logical AND as multiplication.
abstract-algebra ring-theory field-theory terminology
asked Dec 6 '18 at 13:22
GarmekainGarmekain
1,323720
$endgroup$
I find this structure to be useful in my work, but I cannot find any name for it. I think it might be a "commutative division semiring" but I'm not certain because I cannot find any literature on it.
Basically, I'm looking for a name of some set $S$ where $S$ forms a commutative monoid under addition, the nonzero elements of $S$ an abelian group under multiplication, and the distributive law holds. An example is the set ${False,True}$ with logical OR as addition and logical AND as multiplication.
abstract-algebra ring-theory field-theory terminology
abstract-algebra ring-theory field-theory terminology
asked Dec 6 '18 at 13:22
GarmekainGarmekain
1,323720
asked Dec 6 '18 at 13:22
GarmekainGarmekain
1,323720
edited Dec 6 '18 at 13:43
Garmekain
asked Dec 6 '18 at 13:22
GarmekainGarmekain
1,323720
asked Dec 6 '18 at 13:22
GarmekainGarmekain
1,323720
asked Dec 6 '18 at 13:22
GarmekainGarmekain
1,323720
1,323720
add a comment |
add a comment |
2 Answers
2
active
oldest
votes
$begingroup$
I think you are looking for semifields in this sense:
In ring theory, combinatorics, functional analysis, and theoretical computer science (MSC 16Y60), a semifield is a semiring (S,+,·) in which all elements have a multiplicative inverse.
However, I think the passage I copied contains an error when it says all elements. If you look at the references you'll find that they do exclude $0$.
You'll also find, in Golan's book at least, the name "division semiring" used if commutativity of multiplication is required.
The two most useful resources on semirings that I ever found were these:
Golan, Jonathan S. Semirings and their Applications. Springer Science & Business Media, 2013.
and
Gondran, Michel, and Michel Minoux. Graphs, dioids and semirings: new models and algorithms. Vol. 41. Springer Science & Business Media, 2008.
I have never read this but it looks like something to consult:
Glazek, Kazimierz. A guide to the literature on semirings and their applications in mathematics and information sciences: with complete bibliography. Springer Science & Business Media, 2002.
answered Dec 6 '18 at 14:20
rschwiebrschwieb
105k12101246
$endgroup$
$begingroup$
The Wiki article isn't perfect, but you get the drift.
$endgroup$
– rschwieb
Dec 6 '18 at 14:32
$begingroup$
I'll take a look into Golan's book. But it would be the definition you gave for division semiring with an identity, so it is a monoid aditively, instead of just a semiring.
$endgroup$
– Garmekain
Dec 6 '18 at 15:50
1
$begingroup$
@Garmekain I'm pretty sure the tendency is to define semirings on abelian monoids not just semigroups. I think the first two references above require that, anyhow.
$endgroup$
– rschwieb
Dec 6 '18 at 16:15
add a comment |
$begingroup$
For the example you gave, there is a name for that special semiring, and it is called the two-element boolean algebra or boolean semiring.
answered Dec 14 '18 at 20:10
GarmekainGarmekain
1,323720
$endgroup$
add a comment |
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2 Answers
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2 Answers
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$begingroup$
I think you are looking for semifields in this sense:
In ring theory, combinatorics, functional analysis, and theoretical computer science (MSC 16Y60), a semifield is a semiring (S,+,·) in which all elements have a multiplicative inverse.
However, I think the passage I copied contains an error when it says all elements. If you look at the references you'll find that they do exclude $0$.
You'll also find, in Golan's book at least, the name "division semiring" used if commutativity of multiplication is required.
The two most useful resources on semirings that I ever found were these:
Golan, Jonathan S. Semirings and their Applications. Springer Science & Business Media, 2013.
and
Gondran, Michel, and Michel Minoux. Graphs, dioids and semirings: new models and algorithms. Vol. 41. Springer Science & Business Media, 2008.
I have never read this but it looks like something to consult:
Glazek, Kazimierz. A guide to the literature on semirings and their applications in mathematics and information sciences: with complete bibliography. Springer Science & Business Media, 2002.
answered Dec 6 '18 at 14:20
rschwiebrschwieb
105k12101246
$endgroup$
$begingroup$
The Wiki article isn't perfect, but you get the drift.
$endgroup$
– rschwieb
Dec 6 '18 at 14:32
$begingroup$
I'll take a look into Golan's book. But it would be the definition you gave for division semiring with an identity, so it is a monoid aditively, instead of just a semiring.
$endgroup$
– Garmekain
Dec 6 '18 at 15:50
1
$begingroup$
@Garmekain I'm pretty sure the tendency is to define semirings on abelian monoids not just semigroups. I think the first two references above require that, anyhow.
$endgroup$
– rschwieb
Dec 6 '18 at 16:15
add a comment |
$begingroup$
I think you are looking for semifields in this sense:
In ring theory, combinatorics, functional analysis, and theoretical computer science (MSC 16Y60), a semifield is a semiring (S,+,·) in which all elements have a multiplicative inverse.
However, I think the passage I copied contains an error when it says all elements. If you look at the references you'll find that they do exclude $0$.
You'll also find, in Golan's book at least, the name "division semiring" used if commutativity of multiplication is required.
The two most useful resources on semirings that I ever found were these:
Golan, Jonathan S. Semirings and their Applications. Springer Science & Business Media, 2013.
and
Gondran, Michel, and Michel Minoux. Graphs, dioids and semirings: new models and algorithms. Vol. 41. Springer Science & Business Media, 2008.
I have never read this but it looks like something to consult:
Glazek, Kazimierz. A guide to the literature on semirings and their applications in mathematics and information sciences: with complete bibliography. Springer Science & Business Media, 2002.
answered Dec 6 '18 at 14:20
rschwiebrschwieb
105k12101246
$endgroup$
$begingroup$
The Wiki article isn't perfect, but you get the drift.
$endgroup$
– rschwieb
Dec 6 '18 at 14:32
$begingroup$
I'll take a look into Golan's book. But it would be the definition you gave for division semiring with an identity, so it is a monoid aditively, instead of just a semiring.
$endgroup$
– Garmekain
Dec 6 '18 at 15:50
1
$begingroup$
@Garmekain I'm pretty sure the tendency is to define semirings on abelian monoids not just semigroups. I think the first two references above require that, anyhow.
$endgroup$
– rschwieb
Dec 6 '18 at 16:15
add a comment |
$begingroup$
I think you are looking for semifields in this sense:
In ring theory, combinatorics, functional analysis, and theoretical computer science (MSC 16Y60), a semifield is a semiring (S,+,·) in which all elements have a multiplicative inverse.
However, I think the passage I copied contains an error when it says all elements. If you look at the references you'll find that they do exclude $0$.
You'll also find, in Golan's book at least, the name "division semiring" used if commutativity of multiplication is required.
The two most useful resources on semirings that I ever found were these:
Golan, Jonathan S. Semirings and their Applications. Springer Science & Business Media, 2013.
and
Gondran, Michel, and Michel Minoux. Graphs, dioids and semirings: new models and algorithms. Vol. 41. Springer Science & Business Media, 2008.
I have never read this but it looks like something to consult:
Glazek, Kazimierz. A guide to the literature on semirings and their applications in mathematics and information sciences: with complete bibliography. Springer Science & Business Media, 2002.
answered Dec 6 '18 at 14:20
rschwiebrschwieb
105k12101246
$endgroup$
I think you are looking for semifields in this sense:
In ring theory, combinatorics, functional analysis, and theoretical computer science (MSC 16Y60), a semifield is a semiring (S,+,·) in which all elements have a multiplicative inverse.
However, I think the passage I copied contains an error when it says all elements. If you look at the references you'll find that they do exclude $0$.
You'll also find, in Golan's book at least, the name "division semiring" used if commutativity of multiplication is required.
The two most useful resources on semirings that I ever found were these:
Golan, Jonathan S. Semirings and their Applications. Springer Science & Business Media, 2013.
and
Gondran, Michel, and Michel Minoux. Graphs, dioids and semirings: new models and algorithms. Vol. 41. Springer Science & Business Media, 2008.
I have never read this but it looks like something to consult:
Glazek, Kazimierz. A guide to the literature on semirings and their applications in mathematics and information sciences: with complete bibliography. Springer Science & Business Media, 2002.
answered Dec 6 '18 at 14:20
rschwiebrschwieb
105k12101246
answered Dec 6 '18 at 14:20
rschwiebrschwieb
105k12101246
answered Dec 6 '18 at 14:20
rschwiebrschwieb
105k12101246
answered Dec 6 '18 at 14:20
rschwiebrschwieb
105k12101246
105k12101246
$begingroup$
The Wiki article isn't perfect, but you get the drift.
$endgroup$
– rschwieb
Dec 6 '18 at 14:32
$begingroup$
I'll take a look into Golan's book. But it would be the definition you gave for division semiring with an identity, so it is a monoid aditively, instead of just a semiring.
$endgroup$
– Garmekain
Dec 6 '18 at 15:50
1
$begingroup$
@Garmekain I'm pretty sure the tendency is to define semirings on abelian monoids not just semigroups. I think the first two references above require that, anyhow.
$endgroup$
– rschwieb
Dec 6 '18 at 16:15
add a comment |
$begingroup$
The Wiki article isn't perfect, but you get the drift.
$endgroup$
– rschwieb
Dec 6 '18 at 14:32
$begingroup$
I'll take a look into Golan's book. But it would be the definition you gave for division semiring with an identity, so it is a monoid aditively, instead of just a semiring.
$endgroup$
– Garmekain
Dec 6 '18 at 15:50
1
$begingroup$
@Garmekain I'm pretty sure the tendency is to define semirings on abelian monoids not just semigroups. I think the first two references above require that, anyhow.
$endgroup$
– rschwieb
Dec 6 '18 at 16:15
$begingroup$
The Wiki article isn't perfect, but you get the drift.
$endgroup$
– rschwieb
Dec 6 '18 at 14:32
$begingroup$
The Wiki article isn't perfect, but you get the drift.
$endgroup$
– rschwieb
Dec 6 '18 at 14:32
$begingroup$
I'll take a look into Golan's book. But it would be the definition you gave for division semiring with an identity, so it is a monoid aditively, instead of just a semiring.
$endgroup$
– Garmekain
Dec 6 '18 at 15:50
$begingroup$
I'll take a look into Golan's book. But it would be the definition you gave for division semiring with an identity, so it is a monoid aditively, instead of just a semiring.
$endgroup$
– Garmekain
Dec 6 '18 at 15:50
1
1
$begingroup$
@Garmekain I'm pretty sure the tendency is to define semirings on abelian monoids not just semigroups. I think the first two references above require that, anyhow.
$endgroup$
– rschwieb
Dec 6 '18 at 16:15
$begingroup$
@Garmekain I'm pretty sure the tendency is to define semirings on abelian monoids not just semigroups. I think the first two references above require that, anyhow.
$endgroup$
– rschwieb
Dec 6 '18 at 16:15
add a comment |
$begingroup$
For the example you gave, there is a name for that special semiring, and it is called the two-element boolean algebra or boolean semiring.
answered Dec 14 '18 at 20:10
GarmekainGarmekain
1,323720
$endgroup$
add a comment |
$begingroup$
For the example you gave, there is a name for that special semiring, and it is called the two-element boolean algebra or boolean semiring.
answered Dec 14 '18 at 20:10
GarmekainGarmekain
1,323720
$endgroup$
add a comment |
$begingroup$
For the example you gave, there is a name for that special semiring, and it is called the two-element boolean algebra or boolean semiring.
answered Dec 14 '18 at 20:10
GarmekainGarmekain
1,323720
$endgroup$
For the example you gave, there is a name for that special semiring, and it is called the two-element boolean algebra or boolean semiring.
answered Dec 14 '18 at 20:10
GarmekainGarmekain
1,323720
answered Dec 14 '18 at 20:10
GarmekainGarmekain
1,323720
answered Dec 14 '18 at 20:10
GarmekainGarmekain
1,323720
answered Dec 14 '18 at 20:10
GarmekainGarmekain
1,323720
1,323720
add a comment |
add a comment |
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