<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% --> <!-- %% --> <!-- %A addmagma.xml GAP documentation Thomas Breuer --> <!-- %% --> <!-- %% --> <!-- %Y (C) 1998 School Math and Comp. Sci., University of St Andrews, Scotland --> <!-- %Y Copyright (C) 2002 The GAP Group --> <!-- %% -->
<Chapter Label="Additive Magmas">
<Heading>Additive Magmas</Heading>
This chapter deals with domains that are closed under addition <C>+</C>,
which are called <E>near-additive magmas</E> in &GAP;.
Together with the domains closed under multiplication <C>*</C>
(see <Ref Chap="Magmas"/>),
they are the basic algebraic structures.
In many cases, the addition is commutative
(see <Ref Prop="IsAdditivelyCommutative"/>),
the domain is called an <E>additive magma</E> then.
Every module (see <Ref Chap="Modules"/>),
vector space (see <Ref Chap="Vector Spaces"/>),
ring (see <Ref Chap="Rings"/>),
or field (see <Ref Chap="Fields and Division Rings"/>)
is an additive magma.
In the cases of all <E>(near-)additive magma-with-zero</E> or
<E>(near-)additive magma-with-inverses</E>,
additional additive structure is present
(see <Ref Sect="(Near-)Additive Magma Categories"/>).
<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<Section Label="(Near-)Additive Magma Categories">
<Heading>(Near-)Additive Magma Categories</Heading>
<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<Section Label="(Near-)Additive Magma Generation">
<Heading>(Near-)Additive Magma Generation</Heading>
This section describes
functions that create additive magmas from generators
(see <Ref Func="NearAdditiveMagma"/>,
<Ref Func="NearAdditiveMagmaWithZero"/>,
<Ref Func="NearAdditiveGroup"/>),
the underlying operations for which methods can be installed
(see <Ref Oper="NearAdditiveMagmaByGenerators"/>,
<Ref Oper="NearAdditiveMagmaWithZeroByGenerators"/>,
<Ref Oper="NearAdditiveGroupByGenerators"/>)
and functions for forming additive submagmas
(see <Ref Func="SubnearAdditiveMagma"/>,
<Ref Func="SubnearAdditiveMagmaWithZero"/>,
<Ref Func="SubnearAdditiveGroup"/>).
<!-- %The following functions can be used to regard a collection as an additive --> <!-- %magma. --> <!-- % --> <!-- %Declaration{AsAdditiveMagma} --> <!-- %Declaration{AsSubadditiveMagma} -->
</Section>
<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<Section Label="Attributes and Properties for (Near-)Additive Magmas">
<Heading>Attributes and Properties for (Near-)Additive Magmas</Heading>
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