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‎_config.yml

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remote_theme: pages-themes/[email protected]
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plugins:
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- jekyll-remote-theme # add this line to the plugins list if you already have one

‎_sources/api/AbelianGroup.rst.txt

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.. index:: AbelianGroup
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.. _l4162656c69616e47726f7570:
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AbelianGroup
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============
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:viewcode:`catdef.spad line 1 <catdef.spad#L1>` :editcode:`[edit on github] <catdef.spad#L1>`
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The class of abelian groups, i.e. additive monoids where each element has an additive inverse.
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.. _l4162656c69616e47726f7570-30:
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\ :ref:`0 <l4162656c69616e4d6f6e6f6964-30>`\ \: %
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from \ :ref:`AbelianMonoid <l4162656c69616e4d6f6e6f6964>`\
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.. _l4162656c69616e47726f7570-2a:
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.. index::
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pair: AbelianGroup; *
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:spadfun:`\*`\: (\ :ref:`Integer <l496e7465676572>`\ , %) -> %
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\ ``n*x``\ is the product of \ ``x``\ by the integer \ ``n``\ .
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\ :ref:`\* <l4162656c69616e4d6f6e6f6964-2a>`\ \: (\ :ref:`NonNegativeInteger <l4e6f6e4e65676174697665496e7465676572>`\ , %) -> %
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from \ :ref:`AbelianMonoid <l4162656c69616e4d6f6e6f6964>`\
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\ :ref:`\* <l4162656c69616e53656d6947726f7570-2a>`\ \: (\ :ref:`PositiveInteger <l506f736974697665496e7465676572>`\ , %) -> %
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from \ :ref:`AbelianSemiGroup <l4162656c69616e53656d6947726f7570>`\
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.. _l4162656c69616e47726f7570-2b:
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\ :ref:`+ <l4162656c69616e53656d6947726f7570-2b>`\ \: (%, %) -> %
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from \ :ref:`AbelianSemiGroup <l4162656c69616e53656d6947726f7570>`\
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.. _l4162656c69616e47726f7570-2d:
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.. index::
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pair: AbelianGroup; -
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:spadfun:`-`\: % -> %
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\ ``-x``\ is the additive inverse of \ ``x``\ .
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.. index::
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pair: AbelianGroup; -
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:spadfun:`-`\: (%, %) -> %
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\ ``x-y``\ is the difference of \ ``x``\ and \ ``y``\ i.e. \ ``x + (-y)``\ .
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.. _l4162656c69616e47726f7570-3d:
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\ :ref:`= <l426173696354797065-3d>`\ \: (%, %) -> \ :ref:`Boolean <l426f6f6c65616e>`\
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from \ :ref:`BasicType <l426173696354797065>`\
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.. _l4162656c69616e47726f7570-7e3d:
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\ :ref:`~= <l426173696354797065-7e3d>`\ \: (%, %) -> \ :ref:`Boolean <l426f6f6c65616e>`\
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from \ :ref:`BasicType <l426173696354797065>`\
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.. _l4162656c69616e47726f7570-636f65726365:
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\ :ref:`coerce <l436f65726369626c65546f-636f65726365>`\ \: % -> \ :ref:`OutputForm <l4f7574707574466f726d>`\
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from \ :ref:`CoercibleTo <l436f65726369626c65546f>`\ \ :ref:`OutputForm <l4f7574707574466f726d>`\
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.. _l4162656c69616e47726f7570-6c61746578:
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\ :ref:`latex <l53657443617465676f7279-6c61746578>`\ \: % -> \ :ref:`String <l537472696e67>`\
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from \ :ref:`SetCategory <l53657443617465676f7279>`\
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.. _l4162656c69616e47726f7570-6f70706f736974653f:
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\ :ref:`opposite? <l4162656c69616e4d6f6e6f6964-6f70706f736974653f>`\ \: (%, %) -> \ :ref:`Boolean <l426f6f6c65616e>`\
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from \ :ref:`AbelianMonoid <l4162656c69616e4d6f6e6f6964>`\
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.. _l4162656c69616e47726f7570-73616d706c65:
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\ :ref:`sample <l4162656c69616e4d6f6e6f6964-73616d706c65>`\ \: %
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from \ :ref:`AbelianMonoid <l4162656c69616e4d6f6e6f6964>`\
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.. _l4162656c69616e47726f7570-7375627472616374496643616e:
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\ :ref:`subtractIfCan <l43616e63656c6c6174696f6e4162656c69616e4d6f6e6f6964-7375627472616374496643616e>`\ \: (%, %) -> Union(%, failed)
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from \ :ref:`CancellationAbelianMonoid <l43616e63656c6c6174696f6e4162656c69616e4d6f6e6f6964>`\
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.. _l4162656c69616e47726f7570-7a65726f3f:
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\ :ref:`zero? <l4162656c69616e4d6f6e6f6964-7a65726f3f>`\ \: % -> \ :ref:`Boolean <l426f6f6c65616e>`\
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from \ :ref:`AbelianMonoid <l4162656c69616e4d6f6e6f6964>`\
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\ :ref:`AbelianMonoid <l4162656c69616e4d6f6e6f6964>`\
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\ :ref:`AbelianSemiGroup <l4162656c69616e53656d6947726f7570>`\
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\ :ref:`BasicType <l426173696354797065>`\
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\ :ref:`CancellationAbelianMonoid <l43616e63656c6c6174696f6e4162656c69616e4d6f6e6f6964>`\
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\ :ref:`CoercibleTo <l436f65726369626c65546f>`\ \ :ref:`OutputForm <l4f7574707574466f726d>`\
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\ :ref:`SetCategory <l53657443617465676f7279>`\
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‎_sources/api/AbelianMonoid.rst.txt

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.. index:: AbelianMonoid
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.. _l4162656c69616e4d6f6e6f6964:
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AbelianMonoid
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=============
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:viewcode:`catdef.spad line 37 <catdef.spad#L37>` :editcode:`[edit on github] <catdef.spad#L37>`
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The class of multiplicative monoids, i.e. semigroups with an additive identity element.
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.. _l4162656c69616e4d6f6e6f6964-30:
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.. index::
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pair: AbelianMonoid; 0
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:spadfun:`0`\: %
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0 is the additive identity element.
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.. _l4162656c69616e4d6f6e6f6964-2a:
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.. index::
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pair: AbelianMonoid; *
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:spadfun:`\*`\: (\ :ref:`NonNegativeInteger <l4e6f6e4e65676174697665496e7465676572>`\ , %) -> %
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\ ``n * x``\ is left-multiplication by a non negative integer
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\ :ref:`\* <l4162656c69616e53656d6947726f7570-2a>`\ \: (\ :ref:`PositiveInteger <l506f736974697665496e7465676572>`\ , %) -> %
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from \ :ref:`AbelianSemiGroup <l4162656c69616e53656d6947726f7570>`\
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.. _l4162656c69616e4d6f6e6f6964-2b:
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\ :ref:`+ <l4162656c69616e53656d6947726f7570-2b>`\ \: (%, %) -> %
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from \ :ref:`AbelianSemiGroup <l4162656c69616e53656d6947726f7570>`\
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.. _l4162656c69616e4d6f6e6f6964-3d:
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\ :ref:`= <l426173696354797065-3d>`\ \: (%, %) -> \ :ref:`Boolean <l426f6f6c65616e>`\
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from \ :ref:`BasicType <l426173696354797065>`\
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.. _l4162656c69616e4d6f6e6f6964-7e3d:
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\ :ref:`~= <l426173696354797065-7e3d>`\ \: (%, %) -> \ :ref:`Boolean <l426f6f6c65616e>`\
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from \ :ref:`BasicType <l426173696354797065>`\
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.. _l4162656c69616e4d6f6e6f6964-636f65726365:
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\ :ref:`coerce <l436f65726369626c65546f-636f65726365>`\ \: % -> \ :ref:`OutputForm <l4f7574707574466f726d>`\
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from \ :ref:`CoercibleTo <l436f65726369626c65546f>`\ \ :ref:`OutputForm <l4f7574707574466f726d>`\
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.. _l4162656c69616e4d6f6e6f6964-6c61746578:
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\ :ref:`latex <l53657443617465676f7279-6c61746578>`\ \: % -> \ :ref:`String <l537472696e67>`\
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from \ :ref:`SetCategory <l53657443617465676f7279>`\
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.. _l4162656c69616e4d6f6e6f6964-6f70706f736974653f:
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.. index::
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pair: AbelianMonoid; opposite?
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:spadfun:`opposite?`\: (%, %) -> \ :ref:`Boolean <l426f6f6c65616e>`\
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\ ``opposite?(x,y)``\ holds if the sum of \ ``x``\ and \ ``y``\ is \ ``0``\ .
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.. _l4162656c69616e4d6f6e6f6964-73616d706c65:
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.. index::
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pair: AbelianMonoid; sample
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:spadfun:`sample`\: %
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\ ``sample yields``\ a value of type \%
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.. _l4162656c69616e4d6f6e6f6964-7a65726f3f:
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.. index::
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pair: AbelianMonoid; zero?
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:spadfun:`zero?`\: % -> \ :ref:`Boolean <l426f6f6c65616e>`\
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\ ``zero?(x)``\ tests if \ ``x``\ is equal to 0.
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\ :ref:`AbelianSemiGroup <l4162656c69616e53656d6947726f7570>`\
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\ :ref:`BasicType <l426173696354797065>`\
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\ :ref:`CoercibleTo <l436f65726369626c65546f>`\ \ :ref:`OutputForm <l4f7574707574466f726d>`\
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\ :ref:`SetCategory <l53657443617465676f7279>`\
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