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Question Number 190241 by astridmei last updated on 30/Mar/23

$$showthata⊛b=a+ab+bisamonoidwhenG=Z$$

Answered by mehdee42 last updated on 30/Mar/23

$$\ast :\mathbb{Z}\times \mathbb{Z}\to \mathbb{Z};a\ast b=a+b+ab$$$$(\mathbb{Z},\ast )ismonoidwhenever$$$$1)\mathrm{\forall }a,b\in \mathbb{Z}\to a\ast b\in \mathbb{Z}$$$$2)\mathrm{\forall }a,b,c\in \mathbb{Z}\to (a\ast b)\ast c=a\ast (b\ast c)$$$$3)\mathrm{\exists }e\in \mathbb{Z};\mathrm{\forall }a\in \mathbb{Z}\to a\ast e=e\ast a=a$$$$\therefore$$$$1)a\ast b=a+b+ab\in \mathbb{Z}✓$$$$2)(a\ast b)\ast c=(a+b+ab)\ast c=a+b+ab+c+(a+b+ab)c=a+b+c+ab+ac+bc+abc$$$$a\ast (b\ast c)=a\ast (b+c+bc)=a+b+c+bc+a(b+c+bc)=a+b+c+ab+ac+bc+abc\Rightarrow \left(2\right)✓$$$$3)\mathrm{\forall }a\in \mathbb{Z}\to a\ast 0=o\ast a=a✓$$$$$$

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