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 !


Nalin Pithwa

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