aarjaneiro/planetmath huggingface dataset:
bicyclic semigroup

bicyclic semigroup

The

bicyclic semigroup

𝒞

⁢

(

p

,

q

)

is the
monoid generated by

{

p

,

q

}

with
the single

relation

p

⁢

q

=

1

.

The elements of

𝒞

⁢

(

p

,

q

)

are all words of the
form

q

n

⁢

p

m

for

m

,

n

≥

0

(with the understanding

p

0

=

q

0

=

1

).
These words are multiplied as follows:

q

n

⁢

p

m

⁢

q

k

⁢

p

l

=

{

q

n

+

k

-

m

⁢

p

l

if

⁢

m

≤

k

,

q

n

⁢

p

l

+

m

-

k

if

⁢

m

≥

k

.

It is apparent that

𝒞

⁢

(

p

,

q

)

is simple, for if

q

n

⁢

p

m

is an element of

𝒞

⁢

(

p

,

q

)

, then

1

=

p

n

⁢

(

q

n

⁢

p

m

)

⁢

q

m

and so

S

1

⁢

q

n

⁢

p

m

⁢

S

1

=

S

.

It is also easy to see that

𝒞

⁢

(

p

,

q

)

is an

inverse semigroup

: the element

q

n

⁢

p

m

has

inverse

q

m

⁢

p

n

.

It is useful to picture some further

properties

of

𝒞

⁢

(

p

,

q

)

by
arranging the elements in a table:

1

p

p

2

p

3

p

4

…

q

q

⁢

p

q

⁢

p

2

q

⁢

p

3

q

⁢

p

4

…

q

2

q

2

⁢

p

q

2

⁢

p

2

q

2

⁢

p

3

q

2

⁢

p

4

…

q

3

q

3

⁢

p

q

3

⁢

p

2

q

3

⁢

p

3

q

3

⁢

p

4

…

q

4

q

4

⁢

p

q

4

⁢

p

2

q

4

⁢

p

3

q

4

⁢

p

4

…

⋮

⋮

⋮

⋮

⋮

⋱

Then the elements below any horizontal line drawn through this
table form a

right ideal

and the elements to the right of any vertical
line form a

left ideal

.
Further, the elements on the diagonal are all

idempotents

and their standard

ordering

is

1

>

q

⁢

p

>

q

2

⁢

p

2

>

q

3

⁢

p

3

>

⋯

.

Title

bicyclic semigroup

Canonical

name

BicyclicSemigroup

Date of creation

2013-03-22 13:09:57

Last modified on

2013-03-22 13:09:57

Owner

mclase (549)

Last modified by

mclase (549)

Numerical id

8

Author

mclase (549)

Entry type

Definition

Classification

msc 20M99

Synonym

bicyclic monoid

Defines

bicyclic

Generated on Fri Feb 9 19:36:15 2018 by

LaTeXML
congruence

congruence

Let

S

be a

semigroup

. An

equivalence relation

∼

defined on

S

is called a

congruence

if it is preserved under the semigroup

operation

. That is, for all

x

,

y

,

z

∈

S

, if

x

∼

y

then

x

⁢

z

∼

y

⁢

z

and

z

⁢

x

∼

z

⁢

y

.

If

∼

satisfies

only

x

∼

y

implies

x

⁢

z

∼

y

⁢

z

(resp.

z

⁢

x

∼

z

⁢

y

) then

∼

is called a

right congruence

(resp.

left congruence

).

Example

.

Suppose

f

:

S

→

T

is a semigroup homomorphism. Define

∼

by

x

∼

y

iff

f

⁢

(

x

)

=

f

⁢

(

y

)

. Then it is easy to see that

∼

is a congruence.

If

∼

is a congruence, defined on a semigroup

S

,
write

[

x

]

for the

equivalence class

of

x

under

∼

.
Then it is easy to see that

[

x

]

⋅

[

y

]

=

[

x

⁢

y

]

is a well-defined operation on the set of equivalence classes,
and that in fact this set becomes a semigroup with this operation.
This semigroup is called the

quotient

of

S

by

∼

and is written

S

/

∼

.

Thus semigroup are related to

homomorphic images

of semigroups in the same way that

normal subgroups

are related to

homomorphic images of groups

. More precisely, in the group case, the congruence is the coset

relation

, rather than the normal subgroup itself.

Title

congruence

Canonical name

Congruence1

Date of creation

2013-03-22 13:01:08

Last modified on

2013-03-22 13:01:08

Owner

mclase (549)

Last modified by

mclase (549)

Numerical id

7

Author

mclase (549)

Entry type

Definition

Classification

msc 20M99

Related topic

Congruences

Related topic

MultiplicativeCongruence

Related topic

CongruenceRelationOnAnAlgebraicSystem

Defines

quotient semigroup

Generated on Fri Feb 9 19:36:17 2018 by

LaTeXML
cyclic semigroup

cyclic semigroup

A

semigroup

which is generated by a single element is called a

cyclic semigroup

.

Let

S

=

⟨

x

⟩

be a cyclic semigroup. Then as a set,

S

=

{

x

n

⁢

∣

n

>

⁢

0

}

.

If all powers of

x

are distinct, then

S

=

{

x

,

x

2

,

x

3

,

…

}

is (countably) infinite.

Otherwise, there is a least integer

n

>

0

such that

x

n

=

x

m

for some

m

<

n

. It is clear then that the elements

x

,

x

2

,

…

,

x

n

-

1

are distinct, but that for any

j

≥

n

, we must have

x

j

=

x

i

for some

i

,

m

≤

i

≤

n

-

1

. So

S

has

n

-

1

elements.

Unlike in the group case, however, there are in general

multiple

non-isomorphic cyclic semigroups with the same number of elements. In fact, there are

t

non-isomorphic cyclic semigroups with

t

elements: these correspond to the different choices of

m

in the above (with

n

=

t

+

1

).

The integer

m

is called the

index

of

S

, and

n

-

m

is called the

period

of

S

.

The elements

K

=

{

x

m

,

x

m

+

1

,

…

,

x

n

-

1

}

are a

subsemigroup

of

S

. In fact,

K

is a

cyclic group

.

A concrete

representation

of the semigroup with index

m

and period

r

as a

semigroup of transformations

can be obtained as follows. Let

X

=

{

1

,

2

,

3

,

…

,

m

+

r

}

. Let

ϕ

=

(

1

2

3

…

m

+

r

-

1

m

+

r

2

3

4

…

m

+

r

r

+

1

)

.

Then

ϕ

generates a subsemigroup

S

of the full semigroup of transformations

𝒯

X

, and

S

is cyclic with index

m

and period

r

.

Title

cyclic semigroup

Canonical name

CyclicSemigroup

Date of creation

2013-03-22 13:07:30

Last modified on

2013-03-22 13:07:30

Owner

mclase (549)

Last modified by

mclase (549)

Numerical id

6

Author

mclase (549)

Entry type

Definition

Classification

msc 20M99

Defines

index

Defines

period

Generated on Fri Feb 9 19:36:19 2018 by

LaTeXML
identity element is unique

identity element is unique

Theorem

.

The

identity element

of a monoid is unique.

Proof.

Let

e

and

e

′

be identity elements of a monoid

(

G

,

⋅

)

.  Since

e

is an identity element, one has

e

⋅

e

′

=

e

′

.  Since

e

′

is an identity element, one has also

e

⋅

e

′

=

e

.  Thus

e

′

=

e

⋅

e

′

=

e

,

i.e. both identity elements are the same (in inferring this result from the two first

equations

, one has used the

symmetry

(

http://planetmath.org/

Symmetric

) and

transitivity

of the equality relation).

Note.

The theorem also proves the uniqueness of e.g. the identity element of a group, the

additive identity

(

http://planetmath.org/Ring

) 0 of a ring or a field, and the

multiplicative identity

(

http://planetmath.org/Ring

) 1 of a field.

Title

identity element is unique

Canonical name

IdentityElementIsUnique

Date of creation

2013-03-22 18:01:20

Last modified on

2013-03-22 18:01:20

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

11

Author

pahio (2872)

Entry type

Theorem

Classification

msc 20M99

Synonym

neutral element is unique

Synonym

uniqueness of identity element

Related topic

Group

Related topic

UniquenessOfInverseForGroups

Related topic

ZeroVectorInAVectorSpaceIsUnique

Related topic

AbsorbingElement

Generated on Fri Feb 9 19:36:30 2018 by

LaTeXML
monoid

monoid

A monoid is a

semigroup

G

which contains an

identity element

; that is, there exists an

element

e

∈

G

such that

e

⋅

a

=

a

⋅

e

=

a

for all

a

∈

G

.

If

e

and

f

are identity elements of a monoid

G

, then

e

=

e

⋅

f

=

f

⋅

e

=

f

, so we may speak of “the” identity element of

G

.

A

monoid homomorphism

from monoids

G

to

H

is a semigroup homomorphism

f

:

G

→

H

such that

f

⁢

(

e

G

)

=

e

H

, where

e

G

,

e

H

are identity elements of

G

and

H

respectively.

Title

monoid

Canonical

name

Monoid

Date of creation

2013-03-22 11:50:15

Last modified on

2013-03-22 11:50:15

Owner

djao (24)

Last modified by

djao (24)

Numerical id

9

Author

djao (24)

Entry type

Definition

Classification

msc 20M99

Classification

msc 34-01

Synonym

homomorphism

Related topic

Semigroup

Defines

monoid homomorphism

Generated on Fri Feb 9 19:36:32 2018 by

LaTeXML
null semigroup

null semigroup

A

left zero semigroup

is a

semigroup

in which every element is a

left zero

element. In other words, it is a set

S

with a product defined as

x

⁢

y

=

x

for all

x

,

y

∈

S

.

A

right zero semigroup

is defined similarly.

Let

S

be a semigroup. Then

S

is a

null semigroup

if it has a zero element and if the product of any two elements is zero. In other words, there is an element

θ

∈

S

such that

x

⁢

y

=

θ

for all

x

,

y

∈

S

.

Title

null semigroup

Canonical name

NullSemigroup

Date of creation

2013-03-22 13:02:22

Last modified on

2013-03-22 13:02:22

Owner

mclase (549)

Last modified by

mclase (549)

Numerical id

4

Author

mclase (549)

Entry type

Definition

Classification

msc 20M99

Related topic

Semigroup

Related topic

ZeroElements

Defines

null semigroup

Defines

left zero semigroup

Defines

right zero semigroup

Generated on Fri Feb 9 19:36:33 2018 by

LaTeXML
rig

rig

A

rig

(

R

,

+

,

⋅

)

is a set

R

together with two

binary operations

+

:

R

2

→

R

:

(

a

,

b

)

↦

a

+

b

and

⋅

:

R

2

→

R

:

(

a

,

b

)

↦

a

b

, such that both

(

R

,

+

)

and

(

R

,

⋅

)

are monoids, where

⋅

distributes over

+

. That is if

{

a

,

b

,

c

,

d

}

⊂

R

then

(

a

+

b

)

⁢

(

c

+

d

)

=

a

⁢

c

+

a

⁢

d

+

b

⁢

c

+

b

⁢

d

. The

natural numbers

with ordinary

addition

and

multiplication

(

𝐍

,

+

,

⋅

)

is a rig.

A rig

(

R

,

+

,

⋅

)

is a ring if

(

R

,

+

)

is a group. The

integers

with ordinary addition and multiplication

(

𝐙

,

+

,

⋅

)

is a ring.

Title

rig

Canonical

name

Rig

Date of creation

2013-03-22 14:44:29

Last modified on

2013-03-22 14:44:29

Owner

HkBst (6197)

Last modified by

HkBst (6197)

Numerical id

8

Author

HkBst (6197)

Entry type

Definition

Classification

msc 20M99

Related topic

semigroup

Related topic

ring

Generated on Fri Feb 9 19:36:36 2018 by

LaTeXML
semigroup

semigroup

A

semigroup

G

is a set together with a

binary operation

⋅

:

G

×

G

⟶

G

which

satisfies

the associative property:

(

a

⋅

b

)

⋅

c

=

a

⋅

(

b

⋅

c

)

for all

a

,

b

,

c

∈

G

.

The set

G

is not required to be nonempty.

Let

G

,

H

be two semigroups. A

semigroup homomorphism

from

G

to

H

is a

function

f

:

G

→

H

such that

f

⁢

(

a

⁢

b

)

=

f

⁢

(

a

)

⁢

f

⁢

(

b

)

.

Title

semigroup

Canonical name

Semigroup

Date of creation

2013-03-22 11:50:08

Last modified on

2013-03-22 11:50:08

Owner

djao (24)

Last modified by

djao (24)

Numerical id

11

Author

djao (24)

Entry type

Definition

Classification

msc 20M99

Synonym

homomorphism

Related topic

groupoid

Related topic

Band2

Related topic

SubmonoidSubsemigroup

Related topic

NullSemigroup

Related topic

ZeroElements

Related topic

Monoid

Defines

semigroup homomorphism

Generated on Fri Feb 9 19:36:39 2018 by

LaTeXML
semigroup with two elements

semigroup with two elements

Perhaps the simplest non-trivial example of a

semigroup

which is not a group is a particular

semigroup with two elements

. The underlying set of this
semigroup is

{

a

,

b

}

and the

operation

is defined as follows:

a

⋅

a

=

a

a

⋅

b

=

b

b

⋅

a

=

b

b

⋅

b

=

b

It is rather easy to check that this operation is

associative

, as it
should be:

a

⋅

(

a

⋅

a

)

=

a

⋅

a

=

a

=

a

⋅

a

=

(

a

⋅

a

)

⋅

a

a

⋅

(

a

⋅

b

)

=

a

⋅

b

=

b

=

a

⋅

b

=

(

a

⋅

a

)

⋅

b

a

⋅

(

b

⋅

b

)

=

a

⋅

b

=

b

=

b

⋅

b

=

(

a

⋅

b

)

⋅

b

b

⋅

(

a

⋅

a

)

=

b

⋅

a

=

b

=

a

⋅

a

=

(

a

⋅

a

)

⋅

a

a

⋅

(

b

⋅

b

)

=

a

⋅

b

=

b

=

b

⋅

b

=

(

a

⋅

b

)

⋅

b

b

⋅

(

a

⋅

b

)

=

b

⋅

b

=

b

=

b

⋅

b

=

(

b

⋅

a

)

⋅

b

b

⋅

(

b

⋅

a

)

=

b

⋅

b

=

b

=

b

⋅

a

=

(

b

⋅

b

)

⋅

a

b

⋅

(

b

⋅

b

)

=

b

⋅

b

=

b

=

b

⋅

b

=

(

b

⋅

b

)

⋅

b

It is worth noting that this semigroup is

commutative

and has an

identity
element

, which is

a

. It is not a group because the

element

b

does
not have an

inverse

. In fact, it is not even a

cancellative semigroup

because we cannot cancel the

b

in the

equation

a

⋅

b

=

b

⋅

b

.

This semigroup also arises in various

contexts

. For

instance

,
if we choose

a

to be the truth value ”true” and

b

to be the truth
value ”false” and the operation

⋅

to be the

logical connective

”and”,
we obtain this semigroup in logic. We may also represent it by matrices
like so:

a

=

(

1

0

0

1

)

b

=

(

1

0

0

0

)

Title

semigroup with two elements

Canonical name

SemigroupWithTwoElements

Date of creation

2013-03-22 16:21:42

Last modified on

2013-03-22 16:21:42

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

11

Author

rspuzio (6075)

Entry type

Example

Classification

msc 20M99

Generated on Fri Feb 9 19:36:41 2018 by

LaTeXML
semilattice

semilattice

A

lower semilattice

is a

partially ordered set

S in which each pair of

elements

has a

greatest lower bound

.

A

upper semilattice

is a partially ordered set S in which each pair of elements has a least

upper bound

.

Note that it is not normally necessary to distinguish lower from upper semilattices, because one may be converted to the other by reversing the

partial order

. It is normal practise to refer to either

structure

as a

semilattice

and it should be clear from the

context

whether greatest lower bounds or least upper bounds exist.

Alternatively, a semilattice can be considered to be a

commutative

band, that is a

semigroup

which is commutative, and in which every element is

idempotent

. In this context, semilattices are important elements of semigroup theory and play a key role in the structure theory of commutative semigroups.

A partially ordered set which is both a lower semilattice and an upper semilattice is a

lattice

.

Title

semilattice

Canonical name

Semilattice

Date of creation

2013-03-22 12:57:23

Last modified on

2013-03-22 12:57:23

Owner

mclase (549)

Last modified by

mclase (549)

Numerical id

6

Author

mclase (549)

Entry type

Definition

Classification

msc 20M99

Classification

msc 06A12

Related topic

Lattice

Related topic

Poset

Related topic

Idempotent2

Related topic

Join

Related topic

Meet

Related topic

CompleteSemilattice

Defines

lower semilattice

Defines

upper semilattice

Generated on Fri Feb 9 19:36:43 2018 by

LaTeXML
subsemigroup,, submonoid,, and subgroup

subsemigroup,, submonoid,, and subgroup

Let

S

be a

semigroup

, and let

T

be a subset of

S

.

T

is a

subsemigroup

of

S

if

T

is

closed under

the operation of

S

; that it if

x

⁢

y

∈

T

for all

x

,

y

∈

T

.

T

is a

submonoid

of

S

if

T

is a subsemigroup, and

T

has an

identity element

.

T

is a

subgroup

of

S

if

T

is a submonoid which is a group.

Note that submonoids and subgroups do not have to have the same identity element as

S

itself (indeed,

S

may not have an identity element). The identity element may be any

idempotent element

of

S

.

Let

e

∈

S

be an idempotent element. Then there is a

maximal

subsemigroup of

S

for which

e

is the

identity

:

e

⁢

S

⁢

e

=

{

e

⁢

x

⁢

e

∣

x

∈

S

}

.

In addition, there is a maximal subgroup for which

e

is the identity:

𝒰

⁢

(

e

⁢

S

⁢

e

)

=

{

x

∈

e

⁢

S

⁢

e

∣

∃

y

∈

e

⁢

S

⁢

e

⁢

st

⁢

x

⁢

y

=

y

⁢

x

=

e

}

.

Subgroups with different identity elements are disjoint. To see this, suppose that

G

and

H

are subgroups of a semigroup

S

with identity elements

e

and

f

respectively, and suppose

x

∈

G

∩

H

.
Then

x

has an

inverse

y

∈

G

, and an inverse

z

∈

H

. We have:

e

=

x

⁢

y

=

f

⁢

x

⁢

y

=

f

⁢

e

=

z

⁢

x

⁢

e

=

z

⁢

x

=

f

.

Thus intersecting subgroups have the same identity element.

Title

subsemigroup,, submonoid,, and subgroup

Canonical name

SubsemigroupSubmonoidAndSubgroup

Date of creation

2013-03-22 13:02:03

Last modified on

2013-03-22 13:02:03

Owner

mclase (549)

Last modified by

mclase (549)

Numerical id

5

Author

mclase (549)

Entry type

Definition

Classification

msc 20M99

Related topic

Semigroup

Related topic

Subgroup

Defines

subsemigroup

Defines

submonoid

Defines

subgroup

Generated on Fri Feb 9 19:36:54 2018 by

LaTeXML
Green’s equivalences

Green’s equivalences

Let

S

be a semigroup. Green’s equivalences are five equivalences (

http://planetmath.org/EquivalenceRelation

) on

S

:

ℒ

,

ℛ

,

ℋ

,

𝒟

,

𝒥

For all

x

,

y

∈

S

,

x

⁢

ℒ

⁢

y

if

S

1

⁢

x

=

S

1

⁢

y

, i.e.

s

⁢

x

=

y

,

t

⁢

y

=

x

for some

s

,

t

∈

S

1

x

⁢

ℛ

⁢

y

if

x

⁢

S

1

=

y

⁢

S

1

, i.e.

x

⁢

s

=

y

,

y

⁢

t

=

x

for some

s

,

t

∈

S

1

x

⁢

𝒥

⁢

y

if

S

1

⁢

x

⁢

S

1

=

S

1

⁢

y

⁢

S

1

, i.e.

s

⁢

x

⁢

t

=

y

,

u

⁢

y

⁢

v

=

x

for some

s

,

t

,

u

,

v

∈

S

1

x

⁢

ℋ

⁢

y

if

x

⁢

ℒ

⁢

y

and

x

⁢

ℛ

⁢

y

, i.e.

ℋ

=

ℒ

∩

ℛ

x

⁢

𝒟

⁢

y

if

∃

z

∈

S

such that

x

⁢

ℒ

⁢

z

and

z

⁢

ℛ

⁢

y

, i.e.

𝒟

=

ℒ

∘

ℛ

It is clear that

ℋ

⊆

ℒ

,

ℋ

⊆

ℛ

,

ℒ

⊆

𝒟

,

ℛ

⊆

𝒟

,

𝒟

⊆

𝒥

These play a fundamental role in understanding the of semigroups.

Title

Green’s equivalences

Canonical

name

GreensEquivalences

Date of creation

2013-03-22 14:23:12

Last modified on

2013-03-22 14:23:12

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

9

Author

mathcam (2727)

Entry type

Definition

Classification

msc 20Mxx

Synonym

Green’s relations

Generated on Fri Feb 9 19:36:56 2018 by

LaTeXML
absorbing element

absorbing element

An

element

ζ

of a

groupoid

(

G

,

*

)

is called an

absorbing element

(in French

un élément absorbant

) for the

operation

“

*

”, if it

satisfies

ζ

*

a

=

a

*

ζ

=

ζ

for all elements

a

of

G

.

Examples

•

The zero

0

is the absorbing element for

multiplication

(or

multiplicatively absorbing

) in every ring

(

R

,

+

,

⋅

)

.

•

The

zero ideal

(

0

)

is absorbing for ideal multiplication (

http://planetmath.org/IdealMultiplicationLaws

).

•

The

zero vector

0

→

is the absorbing element for the vectoral multiplication (

http://planetmath.org/CrossProduct

) “

×

”.

•

The

empty set

∅

is the absorbing element for the

intersection

operation “

∩

” and also for the

Cartesian product

“

×

”.

•

The “

universal set

”

E

is the absorbing element for the union operation “

∪

”:

E

∪

A

=

A

∪

E

=

E

∀

A

⊆

E

.

•

In an

upper semilattice

, an element is absorbing iff it is the top element (

http://planetmath.org/BoundedLattice

). Dually, an element is absorbing iff it is the bottom element (

http://planetmath.org/BoundedLattice

) in a

lower semilattice

.

As the examples give reason to believe, the absorbing element for an operation is always unique.  Indeed, if in to

ζ

we have in

G

another absorbing element

η

, then we must have

η

=

ζ

*

η

=

ζ

.

Because

ζ

*

ζ

=

ζ

,  the absorbing element is

idempotent

.

If a group has an absorbing element, the group is

trivial

(

http://planetmath.org/

Subgroup

).

Title

absorbing element

Canonical name

AbsorbingElement

Date of creation

2013-03-22 15:46:12

Last modified on

2013-03-22 15:46:12

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

20

Author

pahio (2872)

Entry type

Definition

Classification

msc 20N02

Synonym

absorbant

Synonym

absorbing

Related topic

RingOfSets

Related topic

ZeroElements

Related topic

0cdotA0

Related topic

AbsorbingSet

Related topic

IdentityElementIsUnique

Generated on Fri Feb 9 19:36:58 2018 by

LaTeXML
equation

equation

Equation

An

equation

concerns usually elements of a certain set

M

, where one can say if two

elements

are equal.  In the simplest case,

M

has one

binary operation

“

*

” producing as result some elements of

M

, and these can be compared.  Then, an equation in

(

M

,

*

)

is a

proposition

of the form

E

1

=

E

2

,

(1)

where one has

equated

two expressions

E

1

and

E

2

formed with “

*

” of the elements or

indeterminates

of

M

.  We call the expressions

E

1

and

E

2

respectively the

left hand side

and the

right hand side

of the equation (1).

Example.

Let

S

be a set and

2

S

the set of its subsets.  In the

groupoid

(

2

S

,

∖

)

,  where “

∖

” is the

set difference

, we can write the equation

(

A

∖

B

)

∖

B

=

A

∖

B

(which is always true).

Of course,

M

may be equipped with more

operations

or be a module with some ring of multipliers — then an equation (1) may them.

But one need not assume any

algebraic structure

for the set

M

where the expressions

E

1

and

E

2

are values or where they elements.  Such a situation would occur e.g. if one has a

continuous mapping

f

from a

topological space

L

to another

M

; then one can consider an equation

f

⁢

(

x

)

=

y

.

A somewhat case is the equation

dim

⁡

V

=

2

where

V

is a certain or a

vector space

; both elements of the

extended real number

system.

Root of equation

If an equation (1) in

M

one indeterminate, say

x

, then a value of

x

which

satisfies

(1), i.e. makes it true, is called a

root

or a

solution

of the equation.
Especially, if we have a

polynomial equation

f

⁢

(

x

)

=

0

,  we may speak of the or the

x

0

; it is the

multiplicity

of the zero

x

0

of the

polynomial

f

⁢

(

x

)

. A

multiple root

has multiplicity greater than 1.

Example.

The equation

x

2

+

1

=

x

in the system

ℂ

of the

complex numbers

has as its roots the numbers

x

:=

1

±

i

⁢

3

2

,

which, by the way, are the primitive sixth

roots of unity

.  Their multiplicities are 1.

Title

equation

Canonical name

Equation

Date of creation

2013-03-22 15:28:33

Last modified on

2013-03-22 15:28:33

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

30

Author

pahio (2872)

Entry type

Definition

Classification

msc 20N02

Related topic

Equality2

Related topic

AlgebraicEquation

Related topic

DiophantineEquation

Related topic

TrigonometricEquation

Related topic

DifferenceEquation

Related topic

DifferentialEquation

Related topic

IntegralEquation

Related topic

FunctionalEquation

Related topic

HomogeneousEquation

Related topic

ProportionEquation

Related topic

FiniteDifference

Related topic

RecurrenceRelation

Related topic

CharacteristicEquation

Defines

equate

Defines

side

Defines

root

Defines

solution

Defines

root of an equation

Defines

left hand side

Defines

right hand side

Defines

multiplicity of the root

Defines

order of the root

Defines

multiple root

Generated on Fri Feb 9 19:37:00 2018 by

LaTeXML
groupoid

groupoid

A groupoid

G

is a set together with a

binary operation

⋅

:

G

×

G

⟶

G

. The groupoid (or “magma”) is

closed under

the

operation

.

There is also a separate, category-theoretic (

http://planetmath.org/GroupoidCategoryTheoretic

) definition of “groupoid.”

Title

groupoid

Canonical name

Groupoid

Date of creation

2013-03-22 12:16:03

Last modified on

2013-03-22 12:16:03

Owner

akrowne (2)

Last modified by

akrowne (2)

Numerical id

11

Author

akrowne (2)

Entry type

Definition

Classification

msc 20N02

Synonym

magma

Related topic

Semigroup

Related topic

Group

Related topic

LoopAndQuasigroup

Generated on Fri Feb 9 19:37:01 2018 by

LaTeXML