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 be compact surfaces. Let D be a small disc in S, and in . Let be the circles which form the boundaries of D and respectively. Let be the interiors of D and respectively, and glue to by identifying C with . It can be shown that the resulting surface is “independent” up to homeomorphism, of the various choices made in preceding construction. If denote the homeomorphism classes of S and respectively, we define 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 denotes the class of torus, and denotes the class of the projective plane, then every element of M has a unique expression of the form
where n is an integer greater than or equal to 0 and m is zero, one or two. We have .
This shows that there are interesting examples of monoids and that monoids exist in nature.
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