OpenMath Content Dictionary: generic_alg_cats

Canonical URL:

http://www.openmath.org/cd/generic_alg_cats.ocd

CD File:

generic_alg_cats.ocd

CD as XML Encoded OpenMath:

generic_alg_cats.omcd

Defines:

Abelian_group, Abelian_monoid, Abelian_semigroup, Euclidean_domain, field, group, groupoid, integral_domain, monoid, non_commutative_ring, ordered_Abelian_group, ordered_Abelian_monoid, ordered_group, ordered_monoid, ordered_ring, ring, ringoid, semigroup

Date:

2002-06-17

Version:

0

Review Date:

2005-04-01

Status:

experimental

Uses CD:

algebraic_cats, logic1, meta_cats, set1

A CD of generic algebraic categories. This CD holds information relating to the heirarchical sturcture of the algebraic category system.

monoid

Description:

This Symbol represents the generic category of monoid.

Commented Mathematical property (CMP):

A monoid is a groupoid

Formal Mathematical property (FMP):

xml prefix mathml

<OMOBJ xmlns="http://www.openmath.org/OpenMath">
<OMA>
  <OMS cd="meta_cats" name="has"/>
  <OMS cd="generic_alg_cats" name="monoid"/>
  <OMS cd="generic_alg_cats" name="groupoid"/>
</OMA>
</OMOBJ>

has (monoid, groupoid)

$$\mathrm{has}(\mathrm{monoid},\mathrm{groupoid})$$

Signatures:

sts

Abelian_monoid

Description:

This Symbol represents the generic category of Abelian monoid.

Commented Mathematical property (CMP):

An Abelian monoid is a monoid

Formal Mathematical property (FMP):

xml prefix mathml

<OMOBJ xmlns="http://www.openmath.org/OpenMath">
<OMA>
  <OMS cd="meta_cats" name="has"/>
  <OMS cd="generic_alg_cats" name="Abelian_monoid"/>
  <OMS cd="generic_alg_cats" name="monoid"/>
</OMA>
</OMOBJ>

has (Abelian_monoid, monoid)

$$\mathrm{has}(\mathrm{Abelian\_monoid},\mathrm{monoid})$$

Signatures:

sts

ordered_monoid

Description:

This Symbol represents the generic category of ordered monoid.

Commented Mathematical property (CMP):

An ordered monoid is a monoid

Formal Mathematical property (FMP):

xml prefix mathml

<OMOBJ xmlns="http://www.openmath.org/OpenMath">
<OMA>
  <OMS cd="meta_cats" name="has"/>
  <OMS cd="generic_alg_cats" name="ordered_monoid"/>
  <OMS cd="generic_alg_cats" name="monoid"/>
</OMA>
</OMOBJ>

has (ordered_monoid, monoid)

$$\mathrm{has}(\mathrm{ordered\_monoid},\mathrm{monoid})$$

Signatures:

sts

ordered_Abelian_monoid

Description:

This Symbol represents the generic category of ordered Abelian monoid.

Commented Mathematical property (CMP):

An ordered Abelian monoid is an Abelian monoid

Formal Mathematical property (FMP):

xml prefix mathml

<OMOBJ xmlns="http://www.openmath.org/OpenMath">
<OMA>
  <OMS cd="meta_cats" name="has"/>
  <OMS cd="generic_alg_cats" name="ordered_Abelian_monoid"/>
  <OMS cd="generic_alg_cats" name="Abelian_monoid"/>
</OMA>
</OMOBJ>

has (ordered_Abelian_monoid, Abelian_monoid)

$$\mathrm{has}(\mathrm{ordered\_Abelian\_monoid},\mathrm{Abelian\_monoid})$$

Commented Mathematical property (CMP):

An ordered Abelian monoid is an ordered monoid

Formal Mathematical property (FMP):

xml prefix mathml

<OMOBJ xmlns="http://www.openmath.org/OpenMath">
<OMA>
  <OMS cd="meta_cats" name="has"/>
  <OMS cd="generic_alg_cats" name="ordered_Abelian_monoid"/>
  <OMS cd="generic_alg_cats" name="ordered_monoid"/>
</OMA>
</OMOBJ>

has (ordered_Abelian_monoid, ordered_monoid)

$$\mathrm{has}(\mathrm{ordered\_Abelian\_monoid},\mathrm{ordered\_monoid})$$

Signatures:

sts

groupoid

Description:

This Symbol represents the generic category of groupoid.

Signatures:

sts

semigroup

Description:

This Symbol represents the generic category of semigroup.

Commented Mathematical property (CMP):

A semigroup is a groupoid

Formal Mathematical property (FMP):

xml prefix mathml

<OMOBJ xmlns="http://www.openmath.org/OpenMath">
<OMA>
  <OMS cd="meta_cats" name="has"/>
  <OMS cd="generic_alg_cats" name="semigroup"/>
  <OMS cd="generic_alg_cats" name="groupoid"/>
</OMA>
</OMOBJ>

has (semigroup, groupoid)

$$\mathrm{has}(\mathrm{semigroup},\mathrm{groupoid})$$

Signatures:

sts

Abelian_semigroup

Description:

This Symbol represents the generic category of Abelian semigroup.

Commented Mathematical property (CMP):

An Abelian semigroup is a semigroup

Formal Mathematical property (FMP):

xml prefix mathml

<OMOBJ xmlns="http://www.openmath.org/OpenMath">
<OMA>
  <OMS cd="meta_cats" name="has"/>
  <OMS cd="generic_alg_cats" name="Abelian_semigroup"/>
  <OMS cd="generic_alg_cats" name="semigroup"/>
</OMA>
</OMOBJ>

has (Abelian_semigroup, semigroup)

$$\mathrm{has}(\mathrm{Abelian\_semigroup},\mathrm{semigroup})$$

Signatures:

sts

group

Description:

This Symbol represents the generic category of group.

Commented Mathematical property (CMP):

A group is a monoid

Formal Mathematical property (FMP):

xml prefix mathml

<OMOBJ xmlns="http://www.openmath.org/OpenMath">
<OMA>
  <OMS cd="meta_cats" name="has"/>
  <OMS cd="generic_alg_cats" name="group"/>
  <OMS cd="generic_alg_cats" name="monoid"/>
</OMA>
</OMOBJ>

has (group, monoid)

$$\mathrm{has}(\mathrm{group},\mathrm{monoid})$$

Signatures:

sts

ordered_group

Description:

This Symbol represents the generic category of ordered group.

Commented Mathematical property (CMP):

An ordered group is a group

Formal Mathematical property (FMP):

xml prefix mathml

<OMOBJ xmlns="http://www.openmath.org/OpenMath">
<OMA>
  <OMS cd="meta_cats" name="has"/>
  <OMS cd="generic_alg_cats" name="ordered_group"/>
  <OMS cd="generic_alg_cats" name="group"/>
</OMA>
</OMOBJ>

has (ordered_group, group)

$$\mathrm{has}(\mathrm{ordered\_group},\mathrm{group})$$

Signatures:

sts

Abelian_group

Description:

This Symbol represents the generic category of Abelian group.

Commented Mathematical property (CMP):

An Abelian group is a group

Formal Mathematical property (FMP):

xml prefix mathml

<OMOBJ xmlns="http://www.openmath.org/OpenMath">
<OMA>
  <OMS cd="meta_cats" name="has"/>
  <OMS cd="generic_alg_cats" name="Abelian_group"/>
  <OMS cd="generic_alg_cats" name="group"/>
</OMA>
</OMOBJ>

has (Abelian_group, group)

$$\mathrm{has}(\mathrm{Abelian\_group},\mathrm{group})$$

Signatures:

sts

ordered_Abelian_group

Description:

This Symbol represents the generic category of ordered Abelian group.

Commented Mathematical property (CMP):

An ordered Abelian group is an Abelian group

Formal Mathematical property (FMP):

xml prefix mathml

<OMOBJ xmlns="http://www.openmath.org/OpenMath">
    <OMA>
      <OMS cd="meta_cats" name="has"/>
      <OMS cd="generic_alg_cats" name="ordered_Abelian_group"/>
      <OMS cd="generic_alg_cats" name="Abelian_group"/>
    </OMA>
  </OMOBJ>

has (ordered_Abelian_group, Abelian_group)

$$\mathrm{has}(\mathrm{ordered\_Abelian\_group},\mathrm{Abelian\_group})$$

Commented Mathematical property (CMP):

An ordered Abelian group is an ordered group

Formal Mathematical property (FMP):

xml prefix mathml

<OMOBJ xmlns="http://www.openmath.org/OpenMath">
    <OMA>
      <OMS cd="meta_cats" name="has"/>
      <OMS cd="generic_alg_cats" name="ordered_Abelian_group"/>
      <OMS cd="generic_alg_cats" name="ordered_group"/>
    </OMA>
  </OMOBJ>

has (ordered_Abelian_group, ordered_group)

$$\mathrm{has}(\mathrm{ordered\_Abelian\_group},\mathrm{ordered\_group})$$

Signatures:

sts

ringoid

Description:

This symbol represents the generic category of ringoid.

Signatures:

sts

ring

Description:

This Symbol represents the generic category of ring.

Commented Mathematical property (CMP):

A ring is a ringoid

Formal Mathematical property (FMP):

xml prefix mathml

<OMOBJ xmlns="http://www.openmath.org/OpenMath">
    <OMA>
      <OMS cd="meta_cats" name="has"/>
      <OMS cd="generic_alg_cats" name="ring"/>
      <OMS cd="generic_alg_cats" name="ringoid"/>
    </OMA>
  </OMOBJ>

has (ring, ringoid)

$$\mathrm{has}(\mathrm{ring},\mathrm{ringoid})$$

Commented Mathematical property (CMP):

A ring is a group under addition

Formal Mathematical property (FMP):

xml prefix mathml

<OMOBJ xmlns="http://www.openmath.org/OpenMath">
    <OMA>
      <OMS cd="logic1" name="implies"/>
      <OMA>
        <OMS cd="set1" name="in"/>
        <OMV name="R"/>
        <OMS cd="generic_alg_cats" name="ring"/>
      </OMA>
      <OMA>
        <OMS cd="set1" name="in"/>
        <OMA>
          <OMS cd="algebraic_cats" name="group"/>
          <OMA>
            <OMS cd="algebraic_cats" name="ring_set"/>
            <OMV name="R"/>
          </OMA>
          <OMA>
            <OMS cd="algebraic_cats" name="ring_plus"/>
            <OMV name="R"/>
          </OMA>
          <OMA>
            <OMS cd="algebraic_cats" name="ring_zero"/>
            <OMV name="R"/>
          </OMA>
          <OMA>
            <OMS cd="algebraic_cats" name="ring_negative"/>
            <OMV name="R"/>
          </OMA>
        </OMA>
        <OMS cd="generic_alg_cats" name="group"/>
      </OMA>
    </OMA>
  </OMOBJ>

implies (in ( R, ring) , in (group (ring_set ( R) , ring_plus ( R) , ring_zero ( R) , ring_negative ( R) ) , group) )

$$R\in \mathrm{ring}\Rightarrow \mathrm{group}(\mathrm{ring\_set}\left(R\right),\mathrm{ring\_plus}\left(R\right),\mathrm{ring\_zero}\left(R\right),\mathrm{ring\_negative}\left(R\right))\in \mathrm{group}$$

Commented Mathematical property (CMP):

A ring is a semigroup under multiplication

Formal Mathematical property (FMP):

xml prefix mathml

<OMOBJ xmlns="http://www.openmath.org/OpenMath">
    <OMA>
      <OMS cd="logic1" name="implies"/>
      <OMA>
        <OMS cd="set1" name="in"/>
        <OMV name="R"/>
        <OMS cd="generic_alg_cats" name="ring"/>
      </OMA>
      <OMA>
        <OMS cd="set1" name="in"/>
        <OMA>
          <OMS cd="algebraic_cats" name="semigroup"/>
          <OMA>
            <OMS cd="algebraic_cats" name="ring_set"/>
            <OMV name="R"/>
          </OMA>
          <OMA>
            <OMS cd="algebraic_cats" name="ring_times"/>
            <OMV name="R"/>
          </OMA>
        </OMA>
        <OMS cd="generic_alg_cats" name="semigroup"/>
      </OMA>
    </OMA>
  </OMOBJ>

implies (in ( R, ring) , in (semigroup (ring_set ( R) , ring_times ( R) ) , semigroup) )

$$R\in \mathrm{ring}\Rightarrow \mathrm{semigroup}(\mathrm{ring\_set}\left(R\right),\mathrm{ring\_times}\left(R\right))\in \mathrm{semigroup}$$

Signatures:

sts

ordered_ring

Description:

This Symbol represents the generic category of ordered ring.

Commented Mathematical property (CMP):

An ordered ring is a ring

Formal Mathematical property (FMP):

xml prefix mathml

<OMOBJ xmlns="http://www.openmath.org/OpenMath">
    <OMA>
      <OMS cd="meta_cats" name="has"/>
      <OMS cd="generic_alg_cats" name="ordered_ring"/>
      <OMS cd="generic_alg_cats" name="ring"/>
    </OMA>
  </OMOBJ>

has (ordered_ring, ring)

$$\mathrm{has}(\mathrm{ordered\_ring},\mathrm{ring})$$

Signatures:

sts

non_commutative_ring

Description:

This Symbol represents the generic category of non-commutative ring.

Commented Mathematical property (CMP):

A non-commutative ring is a ring

Formal Mathematical property (FMP):

xml prefix mathml

<OMOBJ xmlns="http://www.openmath.org/OpenMath">
    <OMA>
      <OMS cd="meta_cats" name="has"/>
      <OMS cd="generic_alg_cats" name="non_commutative_ring"/>
      <OMS cd="generic_alg_cats" name="ring"/>
    </OMA>
  </OMOBJ>

has (non_commutative_ring, ring)

$$\mathrm{has}(\mathrm{non\_commutative\_ring},\mathrm{ring})$$

Signatures:

sts

Euclidean_domain

Description:

This Symbol represents the generic category of Euclidean domain.

Commented Mathematical property (CMP):

A Euclidean domain is a ring

Formal Mathematical property (FMP):

xml prefix mathml

<OMOBJ xmlns="http://www.openmath.org/OpenMath">
    <OMA>
      <OMS cd="meta_cats" name="has"/>
      <OMS cd="generic_alg_cats" name="Euclidean_domain"/>
      <OMS cd="generic_alg_cats" name="ring"/>
    </OMA>
  </OMOBJ>

has (Euclidean_domain, ring)

$$\mathrm{has}(\mathrm{Euclidean\_domain},\mathrm{ring})$$

Signatures:

sts

field

Description:

This Symbol represents the generic category of field.

Commented Mathematical property (CMP):

A field is an Abelian group under +

Formal Mathematical property (FMP):

xml prefix mathml

<OMOBJ xmlns="http://www.openmath.org/OpenMath">
    <OMA>
      <OMS cd="logic1" name="implies"/>
      <OMA>
        <OMS cd="set1" name="in"/>
        <OMV name="F"/>
        <OMS cd="generic_alg_cats" name="field"/>
      </OMA>
      <OMA>
        <OMS cd="set1" name="in"/>
        <OMA>
          <OMS cd="algebraic_cats" name="Abelian_group"/>
          <OMA>
            <OMS cd="algebraic_cats" name="field_set"/>
            <OMV name="F"/>
          </OMA>
          <OMA>
            <OMS cd="algebraic_cats" name="field_plus"/>
            <OMV name="F"/>
          </OMA>
          <OMA>
            <OMS cd="algebraic_cats" name="field_zero"/>
            <OMV name="F"/>
          </OMA>
          <OMA>
            <OMS cd="algebraic_cats" name="field_negative"/>
            <OMV name="F"/>
          </OMA>
        </OMA>
        <OMS cd="generic_alg_cats" name="Abelian_group"/>
      </OMA>
    </OMA>
  </OMOBJ>

implies (in ( F, field) , in (Abelian_group (field_set ( F) , field_plus ( F) , field_zero ( F) , field_negative ( F) ) , Abelian_group) )

$$F\in \mathrm{field}\Rightarrow \mathrm{Abelian\_group}(\mathrm{field\_set}\left(F\right),\mathrm{field\_plus}\left(F\right),\mathrm{field\_zero}\left(F\right),\mathrm{field\_negative}\left(F\right))\in \mathrm{Abelian\_group}$$

Commented Mathematical property (CMP):

A field is a ring

Formal Mathematical property (FMP):

xml prefix mathml

<OMOBJ xmlns="http://www.openmath.org/OpenMath">
    <OMA>
      <OMS cd="meta_cats" name="has"/>
      <OMS cd="generic_alg_cats" name="field"/>
      <OMS cd="generic_alg_cats" name="ring"/>
    </OMA>
  </OMOBJ>

has (field, ring)

$$\mathrm{has}(\mathrm{field},\mathrm{ring})$$

Signatures:

sts

integral_domain

Description:

This Symbol represents the generic category of integral domain.

Commented Mathematical property (CMP):

An integral domain is a ring

Formal Mathematical property (FMP):

xml prefix mathml

<OMOBJ xmlns="http://www.openmath.org/OpenMath">
    <OMA>
      <OMS cd="meta_cats" name="has"/>
      <OMS cd="generic_alg_cats" name="integral_domain"/>
      <OMS cd="generic_alg_cats" name="ring"/>
    </OMA>
  </OMOBJ>

has (integral_domain, ring)

$$\mathrm{has}(\mathrm{integral\_domain},\mathrm{ring})$$

Signatures:

sts