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<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->

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<!-- %% -->

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<!-- %A  addmagma.msk               GAP documentation                Thomas Breuer -->

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<!-- %% -->

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<!-- %A  @(#)<M>Id: addmagma.msk,v 1.5 2006/03/10 08:55:52 gap Exp </M> -->

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<!-- %% -->

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<!-- %Y  (C) 1998 School Math and Comp. Sci., University of St Andrews, Scotland -->

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<!-- %Y  Copyright (C) 2002 The GAP Group -->

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<!-- %% -->

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<Chapter Label="Additive Magmas">

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<Heading>Additive Magmas</Heading>

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This chapter deals with domains that are closed under addition <C>+</C>,

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which are called <E>near-additive magmas</E> in &GAP;.

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Together with the domains closed under multiplication <C>*</C>

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(see&nbsp;<Ref Chap="Magmas"/>),

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they are the basic algebraic structures.

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In many cases, the addition is commutative

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(see&nbsp;<Ref Func="IsAdditivelyCommutative"/>),

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the domain is called an <E>additive magma</E> then.

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Every module (see&nbsp;<Ref Chap="Modules"/>),

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vector space (see&nbsp;<Ref Chap="Vector Spaces"/>),

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ring (see&nbsp;<Ref Chap="Rings"/>),

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or field (see&nbsp;<Ref Chap="Fields and Division Rings"/>)

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is an additive magma.

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In the cases of all <E>(near-)additive magma-with-zero</E> or

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<E>(near-)additive magma-with-inverses</E>,

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additional additive structure is present

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(see&nbsp;<Ref Sect="(Near-)Additive Magma Categories"/>).

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<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->

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<Section Label="(Near-)Additive Magma Categories">

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<Heading>(Near-)Additive Magma Categories</Heading>

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<#Include Label="IsNearAdditiveMagma">

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<#Include Label="IsNearAdditiveMagmaWithZero">

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<#Include Label="IsNearAdditiveGroup">

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<#Include Label="IsAdditiveMagma">

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<#Include Label="IsAdditiveMagmaWithZero">

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<#Include Label="IsAdditiveGroup">

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</Section>

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<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->

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<Section Label="(Near-)Additive Magma Generation">

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<Heading>(Near-)Additive Magma Generation</Heading>

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This section describes

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functions that create additive magmas from generators

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(see <Ref Func="NearAdditiveMagma"/>,

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<Ref Func="NearAdditiveMagmaWithZero"/>,

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<Ref Func="NearAdditiveGroup"/>),

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the underlying operations for which methods can be installed

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(see <Ref Func="NearAdditiveMagmaByGenerators"/>,

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<Ref Func="NearAdditiveMagmaWithZeroByGenerators"/>,

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<Ref Func="NearAdditiveGroupByGenerators"/>)

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and functions for forming additive submagmas

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(see <Ref Func="SubnearAdditiveMagma"/>,

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<Ref Func="SubnearAdditiveMagmaWithZero"/>,

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<Ref Func="SubnearAdditiveGroup"/>).

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<#Include Label="NearAdditiveMagma">

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<#Include Label="NearAdditiveMagmaWithZero">

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<#Include Label="NearAdditiveGroup">

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<#Include Label="NearAdditiveMagmaByGenerators">

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<#Include Label="NearAdditiveMagmaWithZeroByGenerators">

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<#Include Label="NearAdditiveGroupByGenerators">

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<#Include Label="SubnearAdditiveMagma">

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<#Include Label="SubnearAdditiveMagmaWithZero">

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<#Include Label="SubnearAdditiveGroup">

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<!-- %The following functions can be used to regard a collection as an additive -->

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<!-- %magma. -->

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<!-- % -->

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<!-- %Declaration{AsAdditiveMagma} -->

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<!-- %Declaration{AsSubadditiveMagma} -->

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</Section>

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<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->

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<Section Label="Attributes and Properties for (Near-)Additive Magmas">

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<Heading>Attributes and Properties for (Near-)Additive Magmas</Heading>

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<#Include Label="IsAdditivelyCommutative">

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<#Include Label="GeneratorsOfNearAdditiveMagma">

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<#Include Label="GeneratorsOfNearAdditiveMagmaWithZero">

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<#Include Label="GeneratorsOfNearAdditiveGroup">

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<#Include Label="AdditiveNeutralElement">

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<#Include Label="TrivialSubnearAdditiveMagmaWithZero">

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</Section>

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<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->

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<Section Label="Operations for (Near-)Additive Magmas">

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<Heading>Operations for (Near-)Additive Magmas</Heading>

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<#Include Label="ClosureNearAdditiveGroup">

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<#Include Label="ShowAdditionTable">

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</Section>

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</Chapter>

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<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->

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<!-- %% -->

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<!-- %E -->

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