# A non trivial example of a monoid

Reference : Algebra 3rd Edition, Serge Lang. AWL International Student Edition.

We assume that the reader is familiar with the terminology of elementary topology. Let M be the set of homeomorphism classes of compact (connected) surfaces. We shall define an addition in M. Let $S, S^{'}$ be compact surfaces. Let D be a small disc in S, and $D^{'}$ in $S^{'}$. Let $C, C^{'}$ be the circles which form the boundaries of D and $D^{'}$ respectively. Let $D_{0}, D_{0}^{'}$ be the interiors of D and $D^{'}$ respectively, and glue $S-D_{0}$ to $S^{'}-D_{0}^{'}$ by identifying C with $C^{'}$. It can be shown that the resulting surface is “independent” up to homeomorphism, of the various choices made in preceding construction. If $\sigma, \sigma_{'}$ denote the homeomorphism classes of S and $S^{'}$ respectively, we define $\sigma + \sigma_{'}$ to be the class of the surface obtained by the preceding gluing process. It can be shown that this addition defines a monoid structure on M, whose unit element is the class of the ordinary 2-sphere. Furthermore, if $\tau$ denotes the class of torus, and $\pi$ denotes the class of the projective plane, then every element $\sigma$ of M has a unique expression of the form

$\sigma = n \tau + m\pi$

where n is an integer greater than or equal to 0 and m is zero, one or two. We have $3\pi=\tau+n$.

This shows that there are interesting examples of monoids and that monoids exist in nature.

Hope you enjoyed !

Regards,

Nalin Pithwa

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