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In the following every surface is assumed to be connected. I've read that the commutative monoid of homeomorphism classes of closed surfaces is generated by $P$ (projective plane) and $T$ (torus) subject to the only(!) relation $P^3=PT$. Here the product is given by the connected sum. Now what about the commutative monoid of homeomorphism classes of compact surfaces (with boundary)? Does it also have a nice presentation? I think it is generated by the $P[k]$ and the $T[k]$, where $[k]$ means that $k$ holes have been inserted, and $k$ runs through the natural numbers. What relations do we need? And how do you prove that no others are needed? This is the really interesting part, because I think everyone can just guess the right relations, namely $P^3 [0] = P[0] T[0]$ and $X[n] Y[m] = (XY)[n+m]$ where $X,Y \in \{P,T\}$. Anyway, I'm sure this is well-known, but I don't know where to read. Perhaps it's a simple reduction to the case of closed surfaces?

Another question, which is rather informal: Do you think that it's worth to read the proofs of these classical classifications? I know the importance of the results, but I suspect that the proofs are just technical.

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In the following every surface is assumed to be connected. I've read that the commutative monoid of homeomorphism classes of closed surfaces is generated by $P$ (projective plane) and $T$ (torus) subject to the only(!) relation $P^3=PT$. Here the product is given by the connected sum. Now what about the commutative monoid of homeomorphism classes of surfaces (with boundary)? Does it also have a nice presentation? I think it is generated by the $P[k]$ and the $T[k]$, where $[k]$ means that $k$ holes have been inserted, and $k$ runs through the natural numbers. What relations do we need? And how do you prove that no others are needed? This is the really interesting part, because I think everyone can just guess the right relations, namely $P^3 [0] = P[0] T[0]$ and $X[n] Y[m] = (XY)[n+m]$ where $X,Y \in \{P,T\}$. Anyway, I'm sure this is well-known, but I don't know where to read. Perhaps it's a simple reduction to the case of closed surfaces?

Another question, which is rather informal: Do you think that it's worth to read the proofs of these classical classifications? I know the importance of the results, but I suspect that the proofs are just technical.

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# Presentation of the monoid of surfaces

In the following every surface is assumed to be connected. I've read that the commutative monoid of homeomorphism classes of closed surfaces is generated by $P$ (projective plane) and $T$ (torus) subject to the only(!) relation $P^3=PT$. Here the product is given by the connected sum. Now what about the commutative monoid of homeomorphism classes of surfaces (with boundary)? Does it also have a nice presentation? I think it is generated by the $P[k]$ and the $T[k]$, where $[k]$ means that $k$ holes have been inserted, and $k$ runs through the natural numbers. What relations do we need? And how do you prove that no others are needed? This is the really interesting part, because I think everyone can just guess the right relations, namely $P^3 [0] = P[0] T[0]$ and $X[n] Y[m] = (XY)[n+m]$ where $X,Y \in \{P,T\}$. Anyway, I'm sure this is well-known, but I don't know where to read.

Another question, which is rather informal: Do you think that it's worth to read the proofs of these classical classifications? I know the importance of the results, but I suspect that the proofs are just technical.