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group [2014/04/07 16:43] nikolaj |
group [2015/02/02 19:02] nikolaj |
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| | @#55CCEE: context | @#55CCEE: $G$ | | | @#55CCEE: context | @#55CCEE: $G$ | | ||
| | @#FFBB00: definiendum | @#FFBB00: $ \langle G,* \rangle \in \mathrm{it}$ | | | @#FFBB00: definiendum | @#FFBB00: $ \langle G,* \rangle \in \mathrm{it}$ | | ||
| - | | @#AAFFAA: inclusion | @#AAFFAA: $\langle G,* \rangle \in \mathrm{monoid}(G)$ | | + | | @#AAFFAA: inclusion | @#AAFFAA: $\langle G,* \rangle \in \mathrm{monoid}(G)$ | |
| - | | @#AADDEE: let | @#AADDEE: $e$ s.t. $\forall g.\, e*a=a*e=a$ | | + | | @#AADDEE: let | @#AADDEE: $e$ | |
| + | | @#AADDEE: such that | @#AADDEE: $\forall g.\, e*a=a*e=a$ | | ||
| | @#DDDDDD: range | @#DDDDDD: $g,g^{-1}\in G$ | | | @#DDDDDD: range | @#DDDDDD: $g,g^{-1}\in G$ | | ||
| | @#55EE55: postulate | @#55EE55: $\forall g.\,\exists g^{-1}.\;(g*g^{-1}=g^{-1}*g=e)$ | | | @#55EE55: postulate | @#55EE55: $\forall g.\,\exists g^{-1}.\;(g*g^{-1}=g^{-1}*g=e)$ | | ||
| - | ==== Discussion ==== | + | ----- |
| - | === Alternative definition === | + | === Alternative definitions === |
| Let $\langle G,* \rangle $ be a set $G$ with a binary operation. I'll rewrite the group axioms explicitly in the first order language: | Let $\langle G,* \rangle $ be a set $G$ with a binary operation. I'll rewrite the group axioms explicitly in the first order language: | ||
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| For given $G$, the set $\text{group}(G)$ is the set of all pairs $\langle G,* \rangle$, containing $G$ itself, as well a binary operation which fulfills the group axioms. One generally calls $G$ the group, i.e. the set with respect to which the operation "$*$" is defined. | For given $G$, the set $\text{group}(G)$ is the set of all pairs $\langle G,* \rangle$, containing $G$ itself, as well a binary operation which fulfills the group axioms. One generally calls $G$ the group, i.e. the set with respect to which the operation "$*$" is defined. | ||
| - | ==== Parents ==== | + | ----- |
| === Subset of === | === Subset of === | ||
| [[Monoid]], [[Loop]] | [[Monoid]], [[Loop]] | ||
