<?xml version="1.0"?>
<!--This OMDoc document is generated from an sTeX-encoded one via LaTeXML, you may want to reconsider editing it.-->
<omdoc xmlns:omdoc="http://omdoc.org/ns" xmlns="http://omdoc.org/ns" xmlns:m="http://www.w3.org/1998/Math/MathML" xmlns:stex="http://kwarc.info/ns/sTeX" xmlns:dc="http://purl.org/dc/elements/1.1/" xmlns:cc="http://creativecommons.org/ns" xmlns:om="http://www.openmath.org/OpenMath">
  <div xmlns="http://www.w3.org/1999/xhtml" class="para" id="p1">
    <p style="" class="p" id="p1.p1"><omdoc:oref class="expandable" href="intro.omdoc"/></p>
  </div>
<omdoc:omgroup layout="sectioning" xml:id="sec.math" about="#sec.math" stex:srcref="paper.tex#textrange(from=40;0,to=133;13)"><omdoc:metadata xml:id="sec.math.metadata1" about="#sec.math.metadata1" stex:srcref="paper.tex#textrange(from=40;0,to=45;13)"><dc:title xml:id="sec.math.metadata1.title1" about="#sec.math.metadata1.title1" stex:srcref="paper.tex#textrange(from=40;0,to=133;13)">Mathematical Content</dc:title></omdoc:metadata><omdoc:omgroup layout="sectioning" xml:id="sec.math.omgroup2" about="#sec.math.omgroup2" stex:srcref="paper.tex#textrange(from=41;0,to=45;13)"><omdoc:metadata xml:id="sec.math.omgroup2.metadata1" about="#sec.math.omgroup2.metadata1" stex:srcref="paper.tex#textrange(from=41;0,to=41;27)"><dc:title xml:id="sec.math.omgroup2.metadata1.title1" about="#sec.math.omgroup2.metadata1.title1" stex:srcref="paper.tex#textrange(from=41;0,to=45;13)">Calculus</dc:title></omdoc:metadata>
  <div xmlns="http://www.w3.org/1999/xhtml" class="para" id="sec.math.omgroup2.p1">
    <p style="" class="p" id="sec.math.omgroup2.p1.p1">We present some standard mathematical definitions, here from calculus.
<omdoc:oref class="expandable" href="continuous.omdoc"/><omdoc:oref class="expandable" href="differentiable.omdoc"/></p>
  </div>
</omdoc:omgroup><omdoc:omgroup layout="sectioning" xml:id="sec.matha" about="#sec.matha" stex:srcref="paper.tex#textrange(from=47;0,to=132;13)"><omdoc:metadata xml:id="sec.matha.metadata1" about="#sec.matha.metadata1" stex:srcref="paper.tex#textrange(from=47;0,to=47;67)"><dc:title xml:id="sec.matha.metadata1.title1" about="#sec.matha.metadata1.title1" stex:srcref="paper.tex#textrange(from=47;0,to=132;13)">A Theory Graph for Elementary Algebra</dc:title></omdoc:metadata>
  <div xmlns="http://www.w3.org/1999/xhtml" class="para" id="sec.matha.p1">
    <p style="" class="p" id="sec.matha.p1.p1">Here we show an example for more advanced theory graph manipulations, in particular
imports via morphisms.
</p>
  </div>
<theory xml:id="magma" about="#magma" stex:srcref="paper.tex#textrange(from=51;0,to=60;12)"><imports from="../../../slides/extcds/omstd/arith1.omdoc#arith1"/><imports from="../../../slides/extcds/omstd/relation1.omdoc#relation1"/><omdoc:imports from="../background/functions.omdoc#functions" xml:id="magma.imports1" about="#magma.imports1" stex:srcref="paper.tex#textrange(from=52;0,to=52;48)"/><omdoc:symbol name="magbase" xml:id="magbase.sym" about="#magbase.sym" stex:srcref="paper.tex#textrange(from=53;1,to=53;22)"/><omdoc:notation cd="magma" name="magbase"><omdoc:prototype><om:OMS cd="magma" name="magbase"/></omdoc:prototype><omdoc:rendering><m:mi>G</m:mi></omdoc:rendering></omdoc:notation><omdoc:symbol name="magmaop" xml:id="magmaop.sym" about="#magmaop.sym" stex:srcref="paper.tex#textrange(from=54;0,to=54;41)"/><omdoc:notation cd="magma" name="magmaop"><omdoc:prototype><om:OMS cd="magma" name="magmaop"/></omdoc:prototype><omdoc:rendering><m:mo>&#x2218;</m:mo></omdoc:rendering></omdoc:notation><omdoc:notation cd="magma" name="magmaop"><omdoc:prototype><om:OMA><om:OMS cd="magma" cr="fun" name="magmaop"/><omdoc:expr name="arg1"/><omdoc:expr name="arg2"/></om:OMA></omdoc:prototype><omdoc:rendering><m:mrow><m:mo egroup="fence" fence="true">(</m:mo><render name="arg1"/><m:mo cr="fun">&#x2218;</m:mo><render name="arg2"/><m:mo egroup="fence" fence="true">)</m:mo></m:mrow></omdoc:rendering></omdoc:notation><omdoc:symbol name="magma" xml:id="magma.def.sym" about="#magma.def.sym" stex:srcref="paper.tex#textrange(from=56;0,to=59;17)"/><omdoc:definition for="magma" xml:id="magma.def" about="#magma.def" stex:srcref="paper.tex#textrange(from=56;0,to=59;17)"><omdoc:CMP xml:id="magma.def.CMP1" about="#magma.def.CMP1" stex:srcref="paper.tex#textrange(from=56;0,to=59;17)">
    <p xmlns="http://www.w3.org/1999/xhtml" style="" class="p" id="magma.def.CMP1.p1">A <omdoc:idx xml:id="magma.def.CMP1.p1.idx1" about="#magma.def.CMP1.p1.idx1" stex:srcref="paper.tex#textrange(from=56;68,to=57;18)"><omdoc:idt xml:id="magma.def.CMP1.p1.idx1.idt1" about="#magma.def.CMP1.p1.idx1.idt1" stex:srcref="paper.tex#textrange(from=56;68,to=57;18)"><omdoc:term cd="magma" name="magma" role="definiendum" xml:id="magma.def.CMP1.p1.idx1.idt1.term1" about="#magma.def.CMP1.p1.idx1.idt1.term1" stex:srcref="paper.tex#textrange(from=56;68,to=57;18)">magma</omdoc:term></omdoc:idt><omdoc:ide index="default" xml:id="magma.def.CMP1.p1.idx1.ide2" about="#magma.def.CMP1.p1.idx1.ide2" stex:srcref="paper.tex#textrange(from=56;68,to=57;18)"><omdoc:idp>magma</omdoc:idp></omdoc:ide></omdoc:idx> is a structure <om:OMOBJ><om:OMA><om:OMS cd="functions" name="tup"/><om:OMS cd="magma" name="magbase"/><om:OMS cd="magma" name="magmaop"/></om:OMA></om:OMOBJ>, such that <om:OMOBJ><om:OMS cd="magma" name="magbase"/></om:OMOBJ> is
closed under the operation <om:OMOBJ><om:OMA><om:OMS cd="functions" name="fun"/><om:OMS cd="magma" name="magmaop"/><om:OMA><om:OMS cd="functions" name="cart"/><om:OMS cd="magma" name="magbase"/><om:OMS cd="magma" name="magbase"/></om:OMA><om:OMS cd="magma" name="magbase"/></om:OMA></om:OMOBJ>.</p>
  </omdoc:CMP></omdoc:definition></theory><theory xml:id="semigroup" about="#semigroup" stex:srcref="paper.tex#textrange(from=62;0,to=68;12)"><imports from="../../../slides/extcds/omstd/arith1.omdoc#arith1"/><imports from="../../../slides/extcds/omstd/relation1.omdoc#relation1"/><omdoc:imports from="#magma" xml:id="semigroup.imports1" about="#semigroup.imports1" stex:srcref="paper.tex#textrange(from=63;0,to=63;22)"/><omdoc:symbol name="semigroup" xml:id="semigroup.def.sym" about="#semigroup.def.sym" stex:srcref="paper.tex#textrange(from=64;0,to=67;18)"/><omdoc:definition for="semigroup" xml:id="semigroup.def" about="#semigroup.def" stex:srcref="paper.tex#textrange(from=64;0,to=67;18)"><omdoc:CMP xml:id="semigroup.def.CMP1" about="#semigroup.def.CMP1" stex:srcref="paper.tex#textrange(from=64;0,to=67;18)">
    <p xmlns="http://www.w3.org/1999/xhtml" style="" class="p" id="semigroup.def.CMP1.p1">A <omdoc:term cd="magma" name="magma" xml:id="semigroup.def.CMP1.p1.term1" about="#semigroup.def.CMP1.p1.term1" stex:srcref="paper.tex#textrange(from=64;63,to=65;26)">magma</omdoc:term><om:OMOBJ><om:OMA><om:OMS cd="functions" name="tup"/><om:OMS cd="magma" name="magbase"/><om:OMS cd="magma" name="magmaop"/></om:OMA></om:OMOBJ>, is called a <omdoc:idx xml:id="semigroup.def.CMP1.p1.idx3" about="#semigroup.def.CMP1.p1.idx3" stex:srcref="paper.tex#textrange(from=64;5,to=65;84)"><omdoc:idt xml:id="semigroup.def.CMP1.p1.idx3.idt1" about="#semigroup.def.CMP1.p1.idx3.idt1" stex:srcref="paper.tex#textrange(from=64;5,to=65;84)"><omdoc:term cd="semigroup" name="semigroup" role="definiendum" xml:id="semigroup.def.CMP1.p1.idx3.idt1.term1" about="#semigroup.def.CMP1.p1.idx3.idt1.term1" stex:srcref="paper.tex#textrange(from=64;5,to=65;84)">semigroup</omdoc:term></omdoc:idt><omdoc:ide index="default" xml:id="semigroup.def.CMP1.p1.idx3.ide2" about="#semigroup.def.CMP1.p1.idx3.ide2" stex:srcref="paper.tex#textrange(from=64;5,to=65;84)"><omdoc:idp>semigroup</omdoc:idp></omdoc:ide></omdoc:idx>, iff
<om:OMOBJ><om:OMS cd="magma" name="magmaop"/></om:OMOBJ> is associative.</p>
  </omdoc:CMP></omdoc:definition></theory><theory xml:id="monoid" about="#monoid" stex:srcref="paper.tex#textrange(from=70;0,to=84;12)"><imports from="../../../slides/extcds/omstd/arith1.omdoc#arith1"/><imports from="../../../slides/extcds/omstd/relation1.omdoc#relation1"/><omdoc:imports from="#semigroup" xml:id="monoid.imports1" about="#monoid.imports1" stex:srcref="paper.tex#textrange(from=71;0,to=71;26)"/><omdoc:symbol name="monneut" xml:id="monneut.sym" about="#monneut.sym" stex:srcref="paper.tex#textrange(from=72;1,to=72;22)"/><omdoc:notation cd="monoid" name="monneut"><omdoc:prototype><om:OMS cd="monoid" name="monneut"/></omdoc:prototype><omdoc:rendering><m:mi>e</m:mi></omdoc:rendering></omdoc:notation><omdoc:symbol name="noneut" xml:id="noneut.sym" about="#noneut.sym" stex:srcref="paper.tex#textrange(from=73;1,to=73;27)"/><omdoc:notation cd="monoid" name="noneut"><omdoc:prototype><om:OMA><om:OMS cd="monoid" cr="fun" name="noneut"/><omdoc:expr name="arg1"/></om:OMA></omdoc:prototype><omdoc:rendering><m:msup><render name="arg1"/><m:mo>*</m:mo></m:msup></omdoc:rendering></omdoc:notation><omdoc:symbol name="monoid" xml:id="monoid.def.sym" about="#monoid.def.sym" stex:srcref="paper.tex#textrange(from=74;0,to=78;17)"/><omdoc:symbol name="neutral-element" xml:id="neutral-element.def.sym" about="#neutral-element.def.sym" stex:srcref="paper.tex#textrange(from=74;0,to=78;17)"/><omdoc:definition for="neutral-element monoid" xml:id="monoid.def" about="#monoid.def" stex:srcref="paper.tex#textrange(from=74;0,to=78;17)"><omdoc:CMP xml:id="monoid.def.CMP1" about="#monoid.def.CMP1" stex:srcref="paper.tex#textrange(from=74;0,to=78;17)">
    <p xmlns="http://www.w3.org/1999/xhtml" style="" class="p" id="monoid.def.CMP1.p1">A <omdoc:idx xml:id="monoid.def.CMP1.p1.idx1" about="#monoid.def.CMP1.p1.idx1" stex:srcref="paper.tex#textrange(from=74;63,to=75;19)"><omdoc:idt xml:id="monoid.def.CMP1.p1.idx1.idt1" about="#monoid.def.CMP1.p1.idx1.idt1" stex:srcref="paper.tex#textrange(from=74;63,to=75;19)"><omdoc:term cd="monoid" name="monoid" role="definiendum" xml:id="monoid.def.CMP1.p1.idx1.idt1.term1" about="#monoid.def.CMP1.p1.idx1.idt1.term1" stex:srcref="paper.tex#textrange(from=74;63,to=75;19)">monoid</omdoc:term></omdoc:idt><omdoc:ide index="default" xml:id="monoid.def.CMP1.p1.idx1.ide2" about="#monoid.def.CMP1.p1.idx1.ide2" stex:srcref="paper.tex#textrange(from=74;63,to=75;19)"><omdoc:idp>monoid</omdoc:idp></omdoc:ide></omdoc:idx> is a structure <om:OMOBJ><om:OMA><om:OMS cd="functions" name="tup"/><om:OMS cd="magma" name="magbase"/><om:OMS cd="magma" name="magmaop"/><om:OMS cd="monoid" name="monneut"/></om:OMA></om:OMOBJ>, such that
<om:OMOBJ><om:OMA><om:OMS cd="functions" name="tup"/><om:OMS cd="magma" name="magbase"/><om:OMS cd="magma" name="magmaop"/></om:OMA></om:OMOBJ> is a <omdoc:term cd="semigroup" name="semigroup" xml:id="monoid.def.CMP1.p1.term4" about="#monoid.def.CMP1.p1.term4" stex:srcref="paper.tex#textrange(from=75;20,to=76;65)">semigroup</omdoc:term> and <om:OMOBJ><om:OMS cd="monoid" name="monneut"/></om:OMOBJ> is a
<omdoc:idx xml:id="monoid.def.CMP1.p1.idx6" about="#monoid.def.CMP1.p1.idx6" stex:srcref="paper.tex#textrange(from=76;64,to=77;28)"><omdoc:idt xml:id="monoid.def.CMP1.p1.idx6.idt1" about="#monoid.def.CMP1.p1.idx6.idt1" stex:srcref="paper.tex#textrange(from=76;64,to=77;28)"><omdoc:term cd="monoid" name="neutral-element" role="definiendum" xml:id="monoid.def.CMP1.p1.idx6.idt1.term1" about="#monoid.def.CMP1.p1.idx6.idt1.term1" stex:srcref="paper.tex#textrange(from=76;64,to=77;28)">neutral element</omdoc:term></omdoc:idt><omdoc:ide index="default" xml:id="monoid.def.CMP1.p1.idx6.ide2" about="#monoid.def.CMP1.p1.idx6.ide2" stex:srcref="paper.tex#textrange(from=76;64,to=77;28)"><omdoc:idp>neutral</omdoc:idp><omdoc:idp>element</omdoc:idp></omdoc:ide></omdoc:idx>, i.e. that <om:OMOBJ><om:OMA><om:OMS cd="relation1" name="eq"/><om:OMA><om:OMS cd="magma" name="magmaop"/><om:OMV name="x"/><om:OMS cd="monoid" name="monneut"/></om:OMA><om:OMV name="x"/></om:OMA></om:OMOBJ> for all <om:OMOBJ><om:OMA><om:OMS cd="functions" name="inset"/><om:OMV name="x"/><om:OMS cd="magma" name="magbase"/></om:OMA></om:OMOBJ>.</p>
  </omdoc:CMP></omdoc:definition><omdoc:definition xml:id="noneut.def" about="#noneut.def" stex:srcref="paper.tex#textrange(from=80;0,to=83;17)"><omdoc:CMP xml:id="noneut.def.CMP1" about="#noneut.def.CMP1" stex:srcref="paper.tex#textrange(from=80;0,to=83;17)">
    <p xmlns="http://www.w3.org/1999/xhtml" style="" class="p" id="noneut.def.CMP1.p1">In a monoid <om:OMOBJ><om:OMA><om:OMS cd="functions" name="tup"/><om:OMS cd="magma" name="magbase"/><om:OMS cd="magma" name="magmaop"/><om:OMS cd="monoid" name="monneut"/></om:OMA></om:OMOBJ>, we use denote the set
<om:OMOBJ><om:OMA><om:OMS cd="functions" name="setst"/><om:OMA><om:OMS cd="functions" name="inset"/><om:OMV name="x"/><om:OMV name="S"/></om:OMA><om:OMA><om:OMS cd="relation1" name="neq"/><om:OMV name="x"/><om:OMS cd="monoid" name="monneut"/></om:OMA></om:OMA></om:OMOBJ> with <om:OMOBJ><om:OMA><om:OMS cd="monoid" name="noneut"/><om:OMV name="S"/></om:OMA></om:OMOBJ>.</p>
  </omdoc:CMP></omdoc:definition></theory><theory xml:id="group" about="#group" stex:srcref="paper.tex#textrange(from=86;0,to=96;12)"><imports from="../../../slides/extcds/omstd/arith1.omdoc#arith1"/><imports from="../../../slides/extcds/omstd/relation1.omdoc#relation1"/><omdoc:imports from="#monoid" xml:id="group.imports1" about="#group.imports1" stex:srcref="paper.tex#textrange(from=87;0,to=87;23)"/><omdoc:symbol name="ginvOp" xml:id="ginvOp.sym" about="#ginvOp.sym" stex:srcref="paper.tex#textrange(from=88;1,to=88;21)"/><omdoc:notation cd="group" name="ginvOp"><omdoc:prototype><om:OMS cd="group" name="ginvOp"/></omdoc:prototype><omdoc:rendering><m:mi>i</m:mi></omdoc:rendering></omdoc:notation><omdoc:symbol name="ginv" xml:id="ginv.sym" about="#ginv.sym" stex:srcref="paper.tex#textrange(from=89;1,to=89;39)"/><omdoc:notation cd="group" name="ginv"><omdoc:prototype><om:OMA><om:OMS cd="group" cr="fun" name="ginv"/><omdoc:expr name="arg1"/></om:OMA></omdoc:prototype><omdoc:rendering><m:mrow><m:mo cr="fun">i</m:mo><m:mrow><m:mo fence="true">(</m:mo><render name="arg1" precedence="-1000000"/><m:mo fence="true">)</m:mo></m:mrow></m:mrow></omdoc:rendering></omdoc:notation><omdoc:symbol name="group" xml:id="group.def.sym" about="#group.def.sym" stex:srcref="paper.tex#textrange(from=90;0,to=95;18)"/><omdoc:symbol name="inverse" xml:id="inverse.def.sym" about="#inverse.def.sym" stex:srcref="paper.tex#textrange(from=90;0,to=95;18)"/><omdoc:definition for="group inverse" xml:id="group.def" about="#group.def" stex:srcref="paper.tex#textrange(from=90;0,to=95;18)"><omdoc:CMP xml:id="group.def.CMP1" about="#group.def.CMP1" stex:srcref="paper.tex#textrange(from=90;0,to=95;18)">
    <p xmlns="http://www.w3.org/1999/xhtml" style="" class="p" id="group.def.CMP1.p1">A <omdoc:idx xml:id="group.def.CMP1.p1.idx1" about="#group.def.CMP1.p1.idx1" stex:srcref="paper.tex#textrange(from=90;71,to=91;18)"><omdoc:idt xml:id="group.def.CMP1.p1.idx1.idt1" about="#group.def.CMP1.p1.idx1.idt1" stex:srcref="paper.tex#textrange(from=90;71,to=91;18)"><omdoc:term cd="group" name="group" role="definiendum" xml:id="group.def.CMP1.p1.idx1.idt1.term1" about="#group.def.CMP1.p1.idx1.idt1.term1" stex:srcref="paper.tex#textrange(from=90;71,to=91;18)">group</omdoc:term></omdoc:idt><omdoc:ide index="default" xml:id="group.def.CMP1.p1.idx1.ide2" about="#group.def.CMP1.p1.idx1.ide2" stex:srcref="paper.tex#textrange(from=90;71,to=91;18)"><omdoc:idp>group</omdoc:idp></omdoc:ide></omdoc:idx> is a structure <om:OMOBJ><om:OMA><om:OMS cd="functions" name="tup"/><om:OMS cd="magma" name="magbase"/><om:OMS cd="magma" name="magmaop"/><om:OMS cd="monoid" name="monneut"/><om:OMS cd="group" name="ginvOp"/></om:OMA></om:OMOBJ>, such that
<om:OMOBJ><om:OMA><om:OMS cd="functions" name="tup"/><om:OMS cd="magma" name="magbase"/><om:OMS cd="magma" name="magmaop"/><om:OMS cd="monoid" name="monneut"/></om:OMA></om:OMOBJ> is a <omdoc:term cd="monoid" name="monoid" xml:id="group.def.CMP1.p1.term4" about="#group.def.CMP1.p1.term4" stex:srcref="paper.tex#textrange(from=91;22,to=92;68)">monoid</omdoc:term> and <om:OMOBJ><om:OMS cd="group" name="ginvOp"/></om:OMOBJ> acts as
a <omdoc:idx xml:id="group.def.CMP1.p1.idx6" about="#group.def.CMP1.p1.idx6" stex:srcref="paper.tex#textrange(from=92;52,to=93;20)"><omdoc:idt xml:id="group.def.CMP1.p1.idx6.idt1" about="#group.def.CMP1.p1.idx6.idt1" stex:srcref="paper.tex#textrange(from=92;52,to=93;20)"><omdoc:term cd="group" name="inverse" role="definiendum" xml:id="group.def.CMP1.p1.idx6.idt1.term1" about="#group.def.CMP1.p1.idx6.idt1.term1" stex:srcref="paper.tex#textrange(from=92;52,to=93;20)">inverse</omdoc:term></omdoc:idt><omdoc:ide index="default" xml:id="group.def.CMP1.p1.idx6.ide2" about="#group.def.CMP1.p1.idx6.ide2" stex:srcref="paper.tex#textrange(from=92;52,to=93;20)"><omdoc:idp>inverse</omdoc:idp></omdoc:ide></omdoc:idx>, i.e. that <om:OMOBJ><om:OMA><om:OMS cd="relation1" name="eq"/><om:OMA><om:OMS cd="magma" name="magmaop"/><om:OMV name="x"/><om:OMA><om:OMS cd="group" name="ginv"/><om:OMV name="x"/></om:OMA></om:OMA><om:OMS cd="monoid" name="monneut"/></om:OMA></om:OMOBJ> for all
<om:OMOBJ><om:OMA><om:OMS cd="functions" name="inset"/><om:OMV name="x"/><om:OMS cd="magma" name="magbase"/></om:OMA></om:OMOBJ>.</p>
  </omdoc:CMP></omdoc:definition></theory><theory xml:id="cgroup" about="#cgroup" stex:srcref="paper.tex#textrange(from=98;0,to=104;12)"><imports from="../../../slides/extcds/omstd/arith1.omdoc#arith1"/><imports from="../../../slides/extcds/omstd/relation1.omdoc#relation1"/><omdoc:imports from="#group" xml:id="cgroup.imports1" about="#cgroup.imports1" stex:srcref="paper.tex#textrange(from=99;0,to=99;20)"/><omdoc:symbol name="commutative-group" xml:id="commutative-group.def.sym" about="#commutative-group.def.sym" stex:srcref="paper.tex#textrange(from=100;0,to=103;16)"/><omdoc:definition for="commutative-group" xml:id="cgroup.def" about="#cgroup.def" stex:srcref="paper.tex#textrange(from=100;0,to=103;16)"><omdoc:CMP xml:id="cgroup.def.CMP1" about="#cgroup.def.CMP1" stex:srcref="paper.tex#textrange(from=100;0,to=103;16)">
    <p xmlns="http://www.w3.org/1999/xhtml" style="" class="p" id="cgroup.def.CMP1.p1">We call a <omdoc:term cd="group" name="group" xml:id="cgroup.def.CMP1.p1.term1" about="#cgroup.def.CMP1.p1.term1" stex:srcref="paper.tex#textrange(from=100;47,to=101;32)">group</omdoc:term><om:OMOBJ><om:OMA><om:OMS cd="functions" name="tup"/><om:OMS cd="magma" name="magbase"/><om:OMS cd="magma" name="magmaop"/><om:OMS cd="monoid" name="monneut"/><om:OMS cd="group" name="ginvOp"/></om:OMA></om:OMOBJ> a
<omdoc:idx xml:id="cgroup.def.CMP1.p1.idx3" about="#cgroup.def.CMP1.p1.idx3" stex:srcref="paper.tex#textrange(from=101;34,to=102;28)"><omdoc:idt xml:id="cgroup.def.CMP1.p1.idx3.idt1" about="#cgroup.def.CMP1.p1.idx3.idt1" stex:srcref="paper.tex#textrange(from=101;34,to=102;28)"><omdoc:term cd="cgroup" name="commutative-group" role="definiendum" xml:id="cgroup.def.CMP1.p1.idx3.idt1.term1" about="#cgroup.def.CMP1.p1.idx3.idt1.term1" stex:srcref="paper.tex#textrange(from=101;34,to=102;28)">commutative group</omdoc:term></omdoc:idt><omdoc:ide index="default" xml:id="cgroup.def.CMP1.p1.idx3.ide2" about="#cgroup.def.CMP1.p1.idx3.ide2" stex:srcref="paper.tex#textrange(from=101;34,to=102;28)"><omdoc:idp>commutative</omdoc:idp><omdoc:idp>group</omdoc:idp></omdoc:ide></omdoc:idx>, iff <om:OMOBJ><om:OMS cd="magma" name="magmaop"/></om:OMOBJ> is commutative.</p>
  </omdoc:CMP></omdoc:definition></theory><theory xml:id="ring" about="#ring" stex:srcref="paper.tex#textrange(from=106;0,to=131;12)"><imports from="../../../slides/extcds/omstd/arith1.omdoc#arith1"/><imports from="../../../slides/extcds/omstd/relation1.omdoc#relation1"/><omdoc:symbol name="rbase" xml:id="rbase.sym" about="#rbase.sym" stex:srcref="paper.tex#textrange(from=107;1,to=107;18)"/><omdoc:notation cd="ring" name="rbase"><omdoc:prototype><om:OMS cd="ring" name="rbase"/></omdoc:prototype><omdoc:rendering><m:mi>R</m:mi></omdoc:rendering></omdoc:notation><omdoc:symbol name="rtimes" xml:id="rtimes.sym" about="#rtimes.sym" stex:srcref="paper.tex#textrange(from=108;0,to=108;37)"/><omdoc:notation cd="ring" name="rtimes"><omdoc:prototype><om:OMS cd="ring" name="rtimes"/></omdoc:prototype><omdoc:rendering><m:mo>&#x22C5;</m:mo></omdoc:rendering></omdoc:notation><omdoc:notation cd="ring" name="rtimes"><omdoc:prototype><om:OMA><om:OMS cd="ring" cr="fun" name="rtimes"/><omdoc:expr name="arg1"/><omdoc:expr name="arg2"/></om:OMA></omdoc:prototype><omdoc:rendering><m:mrow><m:mo egroup="fence" fence="true">(</m:mo><render name="arg1"/><m:mo cr="fun">&#x22C5;</m:mo><render name="arg2"/><m:mo egroup="fence" fence="true">)</m:mo></m:mrow></omdoc:rendering></omdoc:notation><omdoc:symbol name="rone" xml:id="rone.sym" about="#rone.sym" stex:srcref="paper.tex#textrange(from=110;1,to=110;17)"/><omdoc:notation cd="ring" name="rone"><omdoc:prototype><om:OMS cd="ring" name="rone"/></omdoc:prototype><omdoc:rendering><m:mn>1</m:mn></omdoc:rendering></omdoc:notation><omdoc:imports from="#monoid" xml:id="ring.imports4" about="#ring.imports4" stex:srcref="paper.tex#textrange(from=111;0,to=115;21)"><omdoc:morphism><omdoc:requation xml:id="ring.imports4.requation1" about="#ring.imports4.requation1" stex:srcref="paper.tex#textrange(from=112;1,to=112;26)">rbase<om:OMOBJ><om:OMS cd="magma" name="magbase"/></om:OMOBJ></omdoc:requation><omdoc:requation xml:id="ring.imports4.requation2" about="#ring.imports4.requation2" stex:srcref="paper.tex#textrange(from=113;1,to=113;31)">rtimesOp<om:OMOBJ><om:OMS cd="magma" name="magmaop"/></om:OMOBJ></omdoc:requation><omdoc:requation xml:id="ring.imports4.requation3" about="#ring.imports4.requation3" stex:srcref="paper.tex#textrange(from=114;1,to=114;25)">rone<om:OMOBJ><om:OMS cd="monoid" name="monneut"/></om:OMOBJ></omdoc:requation></omdoc:morphism></omdoc:imports><omdoc:symbol name="rplus" xml:id="rplus.sym" about="#rplus.sym" stex:srcref="paper.tex#textrange(from=116;0,to=116;31)"/><omdoc:notation cd="ring" name="rplus"><omdoc:prototype><om:OMS cd="ring" name="rplus"/></omdoc:prototype><omdoc:rendering><m:mo>+</m:mo></omdoc:rendering></omdoc:notation><omdoc:notation cd="ring" name="rplus"><omdoc:prototype><om:OMA><om:OMS cd="ring" cr="fun" name="rplus"/><omdoc:expr name="arg1"/><omdoc:expr name="arg2"/></om:OMA></omdoc:prototype><omdoc:rendering><m:mrow><m:mo egroup="fence" fence="true">(</m:mo><render name="arg1"/><m:mo cr="fun">+</m:mo><render name="arg2"/><m:mo egroup="fence" fence="true">)</m:mo></m:mrow></omdoc:rendering></omdoc:notation><omdoc:symbol name="rzero" xml:id="rzero.sym" about="#rzero.sym" stex:srcref="paper.tex#textrange(from=118;1,to=118;18)"/><omdoc:notation cd="ring" name="rzero"><omdoc:prototype><om:OMS cd="ring" name="rzero"/></omdoc:prototype><omdoc:rendering><m:mn>0</m:mn></omdoc:rendering></omdoc:notation><omdoc:symbol name="rminus" xml:id="rminus.sym" about="#rminus.sym" stex:srcref="paper.tex#textrange(from=119;0,to=119;33)"/><omdoc:notation cd="ring" name="rminus"><omdoc:prototype><om:OMS cd="ring" name="rminus"/></omdoc:prototype><omdoc:rendering><m:mo>-</m:mo></omdoc:rendering></omdoc:notation><omdoc:notation cd="ring" name="rminus"><omdoc:prototype><om:OMA><om:OMS cd="ring" cr="fun" name="rminus"/><omdoc:expr name="arg1"/></om:OMA></omdoc:prototype><omdoc:rendering><m:mrow><m:mo cr="fun">-</m:mo><m:mrow><m:mo fence="true">(</m:mo><render name="arg1" precedence="-1000000"/><m:mo fence="true">)</m:mo></m:mrow></m:mrow></omdoc:rendering></omdoc:notation><omdoc:imports from="#cgroup" xml:id="ring.imports8" about="#ring.imports8" stex:srcref="paper.tex#textrange(from=121;0,to=125;21)"><omdoc:morphism><omdoc:requation xml:id="ring.imports8.requation1" about="#ring.imports8.requation1" stex:srcref="paper.tex#textrange(from=122;1,to=122;28)">rplus<om:OMOBJ><om:OMS cd="magma" name="magmaop"/></om:OMOBJ></omdoc:requation><omdoc:requation xml:id="ring.imports8.requation2" about="#ring.imports8.requation2" stex:srcref="paper.tex#textrange(from=123;1,to=123;26)">rzero<om:OMOBJ><om:OMS cd="monoid" name="monneut"/></om:OMOBJ></omdoc:requation><omdoc:requation xml:id="ring.imports8.requation3" about="#ring.imports8.requation3" stex:srcref="paper.tex#textrange(from=124;1,to=124;28)">rminusOp<om:OMOBJ><om:OMS cd="group" name="ginvOp"/></om:OMOBJ></omdoc:requation></omdoc:morphism></omdoc:imports><omdoc:symbol name="ring" xml:id="ring.def.sym" about="#ring.def.sym" stex:srcref="paper.tex#textrange(from=126;1,to=130;16)"/><omdoc:definition for="ring" xml:id="ring.definition10" about="#ring.definition10" stex:srcref="paper.tex#textrange(from=126;1,to=130;16)"><omdoc:CMP xml:id="ring.definition10.CMP1" about="#ring.definition10.CMP1" stex:srcref="paper.tex#textrange(from=126;1,to=130;16)">
    <p xmlns="http://www.w3.org/1999/xhtml" style="" class="p" id="ring.definition10.CMP1.p1">A <omdoc:idx xml:id="ring.definition10.CMP1.p1.idx1" about="#ring.definition10.CMP1.p1.idx1" stex:srcref="paper.tex#textrange(from=126;73,to=127;15)"><omdoc:idt xml:id="ring.definition10.CMP1.p1.idx1.idt1" about="#ring.definition10.CMP1.p1.idx1.idt1" stex:srcref="paper.tex#textrange(from=126;73,to=127;15)"><omdoc:term cd="ring" name="ring" role="definiendum" xml:id="ring.definition10.CMP1.p1.idx1.idt1.term1" about="#ring.definition10.CMP1.p1.idx1.idt1.term1" stex:srcref="paper.tex#textrange(from=126;73,to=127;15)">ring</omdoc:term></omdoc:idt><omdoc:ide index="default" xml:id="ring.definition10.CMP1.p1.idx1.ide2" about="#ring.definition10.CMP1.p1.idx1.ide2" stex:srcref="paper.tex#textrange(from=126;73,to=127;15)"><omdoc:idp>ring</omdoc:idp></omdoc:ide></omdoc:idx> is a structure <om:OMOBJ><om:OMA><om:OMS cd="functions" name="tup"/><om:OMS cd="ring" name="rbase"/><om:OMS cd="ring" name="rplus"/><om:OMS cd="ring" name="rzero"/><om:OMS cd="ring" name="rtimes"/><om:OMS cd="ring" name="rone"/><om:OMS cd="ring" name="rminus"/></om:OMA></om:OMOBJ>,
such that <om:OMOBJ><om:OMA><om:OMS cd="functions" name="tup"/><om:OMA><om:OMS cd="monoid" name="noneut"/><om:OMS cd="ring" name="rbase"/></om:OMA><om:OMS cd="ring" name="rtimes"/><om:OMS cd="ring" name="rone"/></om:OMA></om:OMOBJ> is a monoid and
<om:OMOBJ><om:OMA><om:OMS cd="functions" name="tup"/><om:OMS cd="ring" name="rbase"/><om:OMS cd="ring" name="rplus"/><om:OMS cd="ring" name="rzero"/><om:OMS cd="ring" name="rminus"/></om:OMA></om:OMOBJ> is a commutative group.</p>
  </omdoc:CMP></omdoc:definition></theory></omdoc:omgroup></omdoc:omgroup><omdoc:omgroup layout="sectioning" xml:id="concl" about="#concl" stex:srcref="paper.tex#textrange(from=135;0,to=138;13)"><omdoc:metadata xml:id="concl.metadata1" about="#concl.metadata1" stex:srcref="paper.tex#textrange(from=135;0,to=135;37)"><dc:title xml:id="concl.metadata1.title1" about="#concl.metadata1.title1" stex:srcref="paper.tex#textrange(from=135;0,to=138;13)">Conclusion</dc:title></omdoc:metadata>
  <div xmlns="http://www.w3.org/1999/xhtml" class="para" id="concl.p1">
    <p style="" class="p" id="concl.p1.p1">In this note we have given an example of standard mathematical markup and shown how a a
sTeX collection can be set up for automation.</p>
  </div>
</omdoc:omgroup><omdoc:bibliography files="kwarc"/></omdoc>
