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group [2014/12/18 18:47] nikolaj |
group [2015/04/16 19:17] 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: $\forall g.\,\exists g^{-1}.\;(g*g^{-1}=g^{-1}*g=e)$ | | ||
| + | ----- | ||
| === Alternative definitions === | === 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: | + | == Sharper definitions == |
| + | We could just define left units and left inverses and prove from the group axioms that they are already units and inverses. | ||
| + | |||
| + | == Group axioms explicitly in the first order language == | ||
| + | Let $\langle G,* \rangle $ be a set $G$ with a binary operation. | ||
| 1. $\forall (a,b\in G).\ (a*b\in G)$ | 1. $\forall (a,b\in G).\ (a*b\in G)$ | ||
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| ----- | ----- | ||
| - | |||
| === Subset of === | === Subset of === | ||
| [[Monoid]], [[Loop]] | [[Monoid]], [[Loop]] | ||
