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 (Revision 1)

Review Date:

2017-12-31

Status:

experimental

---

     This document is distributed in the hope that it will be useful, 
     but WITHOUT ANY WARRANTY; without even the implied warranty of 
     MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.

     The copyright holder grants you permission to redistribute this 
     document freely as a verbatim copy. Furthermore, the copyright
     holder permits you to develop any derived work from this document
     provided that the following conditions are met.
       a) The derived work acknowledges the fact that it is derived from
          this document, and maintains a prominent reference in the 
          work to the original source.
       b) The fact that the derived work is not the original OpenMath 
          document is stated prominently in the derived work.  Moreover if
          both this document and the derived work are Content Dictionaries
          then the derived work must include a different CDName element,
          chosen so that it cannot be confused with any works adopted by
          the OpenMath Society.  In particular, if there is a Content 
          Dictionary Group whose name is, for example, `math' containing
          Content Dictionaries named `math1', `math2' etc., then you should 
          not name a derived Content Dictionary `mathN' where N is an integer.
          However you are free to name it `private_mathN' or some such.  This
          is because the names `mathN' may be used by the OpenMath Society
          for future extensions.
       c) The derived work is distributed under terms that allow the
          compilation of derived works, but keep paragraphs a) and b)
          intact.  The simplest way to do this is to distribute the derived
          work under the OpenMath license, but this is not a requirement.
     If you have questions about this license please contact the OpenMath
     society at http://www.openmath.org.

  Author: Bill Naylor

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):

OpenMath XML (source)

<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>

Strict Content MathML

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="meta_cats">has</csymbol>
  <csymbol cd="generic_alg_cats">monoid</csymbol>
  <csymbol cd="generic_alg_cats">groupoid</csymbol>
 </apply>
</math>

Prefix

has (monoid, groupoid)

Popcorn

meta_cats.has(generic_alg_cats.monoid, generic_alg_cats.groupoid)

Rendered Presentation MathML

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

Signatures:

sts

---

[Next: Abelian_monoid] [Last: integral_domain] [Top]

---

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):

OpenMath XML (source)

<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>

Strict Content MathML

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="meta_cats">has</csymbol>
  <csymbol cd="generic_alg_cats">Abelian_monoid</csymbol>
  <csymbol cd="generic_alg_cats">monoid</csymbol>
 </apply>
</math>

Prefix

has (Abelian_monoid, monoid)

Popcorn

meta_cats.has(generic_alg_cats.Abelian_monoid, generic_alg_cats.monoid)

Rendered Presentation MathML

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

Signatures:

sts

---

[Next: ordered_monoid] [Previous: monoid] [Top]

---

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):

OpenMath XML (source)

<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>

Strict Content MathML

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="meta_cats">has</csymbol>
  <csymbol cd="generic_alg_cats">ordered_monoid</csymbol>
  <csymbol cd="generic_alg_cats">monoid</csymbol>
 </apply>
</math>

Prefix

has (ordered_monoid, monoid)

Popcorn

meta_cats.has(generic_alg_cats.ordered_monoid, generic_alg_cats.monoid)

Rendered Presentation MathML

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

Signatures:

sts

---

[Next: ordered_Abelian_monoid] [Previous: Abelian_monoid] [Top]

---

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):

OpenMath XML (source)

<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>

Strict Content MathML

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="meta_cats">has</csymbol>
  <csymbol cd="generic_alg_cats">ordered_Abelian_monoid</csymbol>
  <csymbol cd="generic_alg_cats">Abelian_monoid</csymbol>
 </apply>
</math>

Prefix

has (ordered_Abelian_monoid, Abelian_monoid)

Popcorn

meta_cats.has(generic_alg_cats.ordered_Abelian_monoid, generic_alg_cats.Abelian_monoid)

Rendered Presentation MathML

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

Commented Mathematical property (CMP):

An ordered Abelian monoid is an ordered monoid

Formal Mathematical property (FMP):

OpenMath XML (source)

<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>

Strict Content MathML

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="meta_cats">has</csymbol>
  <csymbol cd="generic_alg_cats">ordered_Abelian_monoid</csymbol>
  <csymbol cd="generic_alg_cats">ordered_monoid</csymbol>
 </apply>
</math>

Prefix

has (ordered_Abelian_monoid, ordered_monoid)

Popcorn

meta_cats.has(generic_alg_cats.ordered_Abelian_monoid, generic_alg_cats.ordered_monoid)

Rendered Presentation MathML

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

Signatures:

sts

---

[Next: groupoid] [Previous: ordered_monoid] [Top]

---

groupoid

Description:

This Symbol represents the generic category of groupoid.

Signatures:

sts

---

[Next: semigroup] [Previous: ordered_Abelian_monoid] [Top]

---

semigroup

Description:

This Symbol represents the generic category of semigroup.

Commented Mathematical property (CMP):

A semigroup is a groupoid

Formal Mathematical property (FMP):

OpenMath XML (source)

<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>

Strict Content MathML

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="meta_cats">has</csymbol>
  <csymbol cd="generic_alg_cats">semigroup</csymbol>
  <csymbol cd="generic_alg_cats">groupoid</csymbol>
 </apply>
</math>

Prefix

has (semigroup, groupoid)

Popcorn

meta_cats.has(generic_alg_cats.semigroup, generic_alg_cats.groupoid)

Rendered Presentation MathML

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

Signatures:

sts

---

[Next: Abelian_semigroup] [Previous: groupoid] [Top]

---

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):

OpenMath XML (source)

<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>

Strict Content MathML

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="meta_cats">has</csymbol>
  <csymbol cd="generic_alg_cats">Abelian_semigroup</csymbol>
  <csymbol cd="generic_alg_cats">semigroup</csymbol>
 </apply>
</math>

Prefix

has (Abelian_semigroup, semigroup)

Popcorn

meta_cats.has(generic_alg_cats.Abelian_semigroup, generic_alg_cats.semigroup)

Rendered Presentation MathML

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

Signatures:

sts

---

[Next: group] [Previous: semigroup] [Top]

---

group

Description:

This Symbol represents the generic category of group.

Commented Mathematical property (CMP):

A group is a monoid

Formal Mathematical property (FMP):

OpenMath XML (source)

<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>

Strict Content MathML

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="meta_cats">has</csymbol>
  <csymbol cd="generic_alg_cats">group</csymbol>
  <csymbol cd="generic_alg_cats">monoid</csymbol>
 </apply>
</math>

Prefix

has (group, monoid)

Popcorn

meta_cats.has(generic_alg_cats.group, generic_alg_cats.monoid)

Rendered Presentation MathML

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

Signatures:

sts

---

[Next: ordered_group] [Previous: Abelian_semigroup] [Top]

---

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):

OpenMath XML (source)

<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>

Strict Content MathML

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="meta_cats">has</csymbol>
  <csymbol cd="generic_alg_cats">ordered_group</csymbol>
  <csymbol cd="generic_alg_cats">group</csymbol>
 </apply>
</math>

Prefix

has (ordered_group, group)

Popcorn

meta_cats.has(generic_alg_cats.ordered_group, generic_alg_cats.group)

Rendered Presentation MathML

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

Signatures:

sts

---

[Next: Abelian_group] [Previous: group] [Top]

---

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):

OpenMath XML (source)

<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>

Strict Content MathML

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="meta_cats">has</csymbol>
  <csymbol cd="generic_alg_cats">Abelian_group</csymbol>
  <csymbol cd="generic_alg_cats">group</csymbol>
 </apply>
</math>

Prefix

has (Abelian_group, group)

Popcorn

meta_cats.has(generic_alg_cats.Abelian_group, generic_alg_cats.group)

Rendered Presentation MathML

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

Signatures:

sts

---

[Next: ordered_Abelian_group] [Previous: ordered_group] [Top]

---

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):

OpenMath XML (source)

  <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>

Strict Content MathML

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="meta_cats">has</csymbol>
  <csymbol cd="generic_alg_cats">ordered_Abelian_group</csymbol>
  <csymbol cd="generic_alg_cats">Abelian_group</csymbol>
 </apply>
</math>

Prefix

has (ordered_Abelian_group, Abelian_group)

Popcorn

meta_cats.has(generic_alg_cats.ordered_Abelian_group, generic_alg_cats.Abelian_group)

Rendered Presentation MathML

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

Commented Mathematical property (CMP):

An ordered Abelian group is an ordered group

Formal Mathematical property (FMP):

OpenMath XML (source)

  <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>

Strict Content MathML

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="meta_cats">has</csymbol>
  <csymbol cd="generic_alg_cats">ordered_Abelian_group</csymbol>
  <csymbol cd="generic_alg_cats">ordered_group</csymbol>
 </apply>
</math>

Prefix

has (ordered_Abelian_group, ordered_group)

Popcorn

meta_cats.has(generic_alg_cats.ordered_Abelian_group, generic_alg_cats.ordered_group)

Rendered Presentation MathML

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

Signatures:

sts

---

[Next: ringoid] [Previous: Abelian_group] [Top]

---

ringoid

Description:

This symbol represents the generic category of ringoid.

Signatures:

sts

---

[Next: ring] [Previous: ordered_Abelian_group] [Top]

---

ring

Description:

This Symbol represents the generic category of ring.

Commented Mathematical property (CMP):

A ring is a ringoid

Formal Mathematical property (FMP):

OpenMath XML (source)

  <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>

Strict Content MathML

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="meta_cats">has</csymbol>
  <csymbol cd="generic_alg_cats">ring</csymbol>
  <csymbol cd="generic_alg_cats">ringoid</csymbol>
 </apply>
</math>

Prefix

has (ring, ringoid)

Popcorn

meta_cats.has(generic_alg_cats.ring, generic_alg_cats.ringoid)

Rendered Presentation MathML

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

Commented Mathematical property (CMP):

A ring is a group under addition

Formal Mathematical property (FMP):

OpenMath XML (source)

  <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>

Strict Content MathML

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="logic1">implies</csymbol>
  <apply><csymbol cd="set1">in</csymbol><ci>R</ci><csymbol cd="generic_alg_cats">ring</csymbol></apply>
  <apply><csymbol cd="set1">in</csymbol>
   <apply><csymbol cd="algebraic_cats">group</csymbol>
    <apply><csymbol cd="algebraic_cats">ring_set</csymbol><ci>R</ci></apply>
    <apply><csymbol cd="algebraic_cats">ring_plus</csymbol><ci>R</ci></apply>
    <apply><csymbol cd="algebraic_cats">ring_zero</csymbol><ci>R</ci></apply>
    <apply><csymbol cd="algebraic_cats">ring_negative</csymbol><ci>R</ci></apply>
   </apply>
   <csymbol cd="generic_alg_cats">group</csymbol>
  </apply>
 </apply>
</math>

Prefix

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

Popcorn

set1.in($R, generic_alg_cats.ring) ==> set1.in(algebraic_cats.group(algebraic_cats.ring_set($R), algebraic_cats.ring_plus($R), algebraic_cats.ring_zero($R), algebraic_cats.ring_negative($R)), generic_alg_cats.group)

Rendered Presentation MathML

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

Commented Mathematical property (CMP):

A ring is a semigroup under multiplication

Formal Mathematical property (FMP):

OpenMath XML (source)

  <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>

Strict Content MathML

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="logic1">implies</csymbol>
  <apply><csymbol cd="set1">in</csymbol><ci>R</ci><csymbol cd="generic_alg_cats">ring</csymbol></apply>
  <apply><csymbol cd="set1">in</csymbol>
   <apply><csymbol cd="algebraic_cats">semigroup</csymbol>
    <apply><csymbol cd="algebraic_cats">ring_set</csymbol><ci>R</ci></apply>
    <apply><csymbol cd="algebraic_cats">ring_times</csymbol><ci>R</ci></apply>
   </apply>
   <csymbol cd="generic_alg_cats">semigroup</csymbol>
  </apply>
 </apply>
</math>

Prefix

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

Popcorn

set1.in($R, generic_alg_cats.ring) ==> set1.in(algebraic_cats.semigroup(algebraic_cats.ring_set($R), algebraic_cats.ring_times($R)), generic_alg_cats.semigroup)

Rendered Presentation MathML

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

Signatures:

sts

---

[Next: ordered_ring] [Previous: ringoid] [Top]

---

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):

OpenMath XML (source)

  <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>

Strict Content MathML

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="meta_cats">has</csymbol>
  <csymbol cd="generic_alg_cats">ordered_ring</csymbol>
  <csymbol cd="generic_alg_cats">ring</csymbol>
 </apply>
</math>

Prefix

has (ordered_ring, ring)

Popcorn

meta_cats.has(generic_alg_cats.ordered_ring, generic_alg_cats.ring)

Rendered Presentation MathML

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

Signatures:

sts

---

[Next: non_commutative_ring] [Previous: ring] [Top]

---

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):

OpenMath XML (source)

  <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>

Strict Content MathML

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="meta_cats">has</csymbol>
  <csymbol cd="generic_alg_cats">non_commutative_ring</csymbol>
  <csymbol cd="generic_alg_cats">ring</csymbol>
 </apply>
</math>

Prefix

has (non_commutative_ring, ring)

Popcorn

meta_cats.has(generic_alg_cats.non_commutative_ring, generic_alg_cats.ring)

Rendered Presentation MathML

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

Signatures:

sts

---

[Next: Euclidean_domain] [Previous: ordered_ring] [Top]

---

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):

OpenMath XML (source)

  <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>

Strict Content MathML

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="meta_cats">has</csymbol>
  <csymbol cd="generic_alg_cats">Euclidean_domain</csymbol>
  <csymbol cd="generic_alg_cats">ring</csymbol>
 </apply>
</math>

Prefix

has (Euclidean_domain, ring)

Popcorn

meta_cats.has(generic_alg_cats.Euclidean_domain, generic_alg_cats.ring)

Rendered Presentation MathML

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

Signatures:

sts

---

[Next: field] [Previous: non_commutative_ring] [Top]

---

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):

OpenMath XML (source)

  <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>

Strict Content MathML

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="logic1">implies</csymbol>
  <apply><csymbol cd="set1">in</csymbol><ci>F</ci><csymbol cd="generic_alg_cats">field</csymbol></apply>
  <apply><csymbol cd="set1">in</csymbol>
   <apply><csymbol cd="algebraic_cats">Abelian_group</csymbol>
    <apply><csymbol cd="algebraic_cats">field_set</csymbol><ci>F</ci></apply>
    <apply><csymbol cd="algebraic_cats">field_plus</csymbol><ci>F</ci></apply>
    <apply><csymbol cd="algebraic_cats">field_zero</csymbol><ci>F</ci></apply>
    <apply><csymbol cd="algebraic_cats">field_negative</csymbol><ci>F</ci></apply>
   </apply>
   <csymbol cd="generic_alg_cats">Abelian_group</csymbol>
  </apply>
 </apply>
</math>

Prefix

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

Popcorn

set1.in($F, generic_alg_cats.field) ==> set1.in(algebraic_cats.Abelian_group(algebraic_cats.field_set($F), algebraic_cats.field_plus($F), algebraic_cats.field_zero($F), algebraic_cats.field_negative($F)), generic_alg_cats.Abelian_group)

Rendered Presentation MathML

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

Commented Mathematical property (CMP):

A field is a ring

Formal Mathematical property (FMP):

OpenMath XML (source)

  <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>

Strict Content MathML

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="meta_cats">has</csymbol>
  <csymbol cd="generic_alg_cats">field</csymbol>
  <csymbol cd="generic_alg_cats">ring</csymbol>
 </apply>
</math>

Prefix

has (field, ring)

Popcorn

meta_cats.has(generic_alg_cats.field, generic_alg_cats.ring)

Rendered Presentation MathML

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

Signatures:

sts

---

[Next: integral_domain] [Previous: Euclidean_domain] [Top]

---

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):

OpenMath XML (source)

  <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>

Strict Content MathML

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="meta_cats">has</csymbol>
  <csymbol cd="generic_alg_cats">integral_domain</csymbol>
  <csymbol cd="generic_alg_cats">ring</csymbol>
 </apply>
</math>

Prefix

has (integral_domain, ring)

Popcorn

meta_cats.has(generic_alg_cats.integral_domain, generic_alg_cats.ring)

Rendered Presentation MathML

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

Signatures:

sts

---

[First: monoid] [Previous: field] [Top]