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I have search out my whole maths book for it..but didn't got this question in my book and I need to learn it for my exam..So if you people can please solve this for me?

$S=\{0,1,2,3\}$. Show that $(S,+)$ is a Semi group where $+$ defines addition Modulo 4?

Waiting for your reply please..

naslundx
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Umair Shah
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1 Answers1

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In short a semigroup is a set $S$ together with some binary operation $\star$ on it that is associative. That means that for every pair of elements $x,y\in S$ there is an element $x\star y\in S$ and the rule $\left(x\star y\right)\star z=x\star\left(y\star z\right)$ is obeyed. An example is $\mathbb{N}=\left\{ 1,2,\dots\right\} $ together with the addition as operation, denoted by $+$. It is a semigroup since rule $\left(n+m\right)+k=n+\left(m+k\right)$ is satisfied. In your case we deal with $S=\left\{ 0,1,2,3\right\} $ and the operation is addition modulo $4$. To show that it is a semigroup it must be verified that the rule of associativity is valid in this situation. That is what you are asked to do.

Caution: check whether this definition of semigroup agrees with the one in your book. It can be that it is also demanded that $S$ contains some specific element $e$ having the property that $e\star s=s\star e=s$ for each $s\in S$. I would speak of a 'monoid' instead of a semigroup, but others don't. If that is the case then the mentioned example $\mathbb{N}=\left\{ 1,2,\dots\right\} $ with addition should be replaced by $\mathbb{N}\cup\{0\}$ and $0$ is the specific element (called identity or unit).

drhab
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  • Thanks...@drhab for explaining this to me...but I am not sure....as normally just as you said the rule is (x⋆y)⋆z=x⋆(y⋆z) and they are asking something very different...so in simple case what can be the solution of this question just in short..? – Umair Shah Mar 24 '14 at 12:48
  • Then what very different (and different from what?) are they asking??? That does not come forward in your question or comments. It is simply enough to say: For integers $n,m,k$ we have $\left(n+m\right)+k=n+\left(m+k\right)$ and consequently $\left(n+m\right)+k\equiv n+\left(m+k\right)$ mod $4$. This proves that $S=\left{ 0,1,2,3\right} $ equipped with addition mod $4$ is a semigroup. – drhab Mar 24 '14 at 13:44
  • So should I write the answer to them in this way as? : For integers n,m,k we have (n+m)+k=n+(m+k) and consequently (n+m)+k≡n+(m+k) mod 4. This proves that S={0,1,2,3} equipped with addition mod 4 is a semigroup – Umair Shah Mar 24 '14 at 13:51
  • If I would have asked you the question and you should have given this answer then I would be completely satisfied. Did you check about what I wrote in my answer under the word 'caution'? – drhab Mar 24 '14 at 13:55
  • Yeah..sure...the problem is that this question is not in my book..may be seems as the course got changed last year..so this question is may be from old book..but I am not sure... – Umair Shah Mar 24 '14 at 14:20