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.error { color:#AA0000; }</style></head><body style="word-wrap: break-word; -webkit-nbsp-mode: space; -webkit-line-break: after-white-space;"><div class="bloop_markdown"><p>Hi,</p>
<p>While they have similarities, they are not the same. Semigroup <code>(S,*)</code> is a set <code>S</code> with associative binary operation <code>*</code>. Monoid is a semigroup that has identity element <code>i</code> such that <code>i * a = a</code> and <code>a * i = a</code>. And group is a monoid in which every element of set has it’s own inverse: <code>a * b = i</code> and <code>b * a = i</code> where <code>b</code> is an inverse of <code>a</code>. </p>
<p>Let’s look at the definition of <code>Monoid</code> in haskell.</p>
<pre><code>class Monoid a where
mempty :: a
-- ^ Identity of 'mappend'
mappend :: a -> a -> a
-- ^ An associative operation
mconcat :: [a] -> a
-- ^ Fold a list using the monoid.
-- For most types, the default definition for 'mconcat' will be
-- used, but the function is included in the class definition so
-- that an optimized version can be provided for specific types.
mconcat = foldr mappend mempty
</code></pre>
<p>So we have a set of <code>a</code>, associative binary operation <code>mappend</code> and identity element <code>mempty</code>. The only difference between <code>Monoid a</code> in haskell and monoid in algebra is that <code>Monoid</code> in haskell has <code>mconcat</code> function in it’s definition. But you can ignore it. </p>
<p>I hope it helps.</p>
<p>Cheers,
d12frosted</p>
<p></p></div><div class="bloop_original_html"><style>body{font-family:Helvetica,Arial;font-size:13px}</style><div id="bloop_customfont" style="font-family:Helvetica,Arial;font-size:13px; color: rgba(0,0,0,1.0); margin: 0px; line-height: auto;"><br></div><p class="airmail_on" style="color:#000;">On March 25, 2015 at 16:03:14, Shishir Srivastava (<a href="mailto:shishir.srivastava@gmail.com">shishir.srivastava@gmail.com</a>) wrote:</p> <blockquote type="cite" class="clean_bq"><span><div><div></div><div>
<title></title>
<div dir="ltr">Hi,
<div><br></div>
<div>Reading about Monoids it seems they derive a lot on the
algebraic structures of 'Groups' ? </div>
<div><br></div>
<div>Is it then correct to assume that Monoids can be used to
represent 'Groups' ?</div>
<div><br></div>
<div>If not are there any standard haskell libraries which
represent algebraic structures such as 'Groups' , 'Fields'
etc.</div>
<div><br></div>
<div>Thanks,</div>
<div>
<div>
<div class="gmail_signature">
<div dir="ltr"><font color="#0B5394"><font style="background-color:rgb(255,255,255)"><font face="georgia, serif">Shishir
Srivastava</font></font><br></font><br></div>
</div>
</div>
</div>
</div>
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