Could anyone tell me what the mathematical structure is called if we replace commutative group by commutative monoid in the definition of linear space?
Also, are there any names for "commutative monoid" structure Banach and Hilbert space-like space?


Thanks for your help.



What is the mathematical structure called if we replace commutative group by commutative monoid in the definition of linear space?
Let me expand my comments in an short answer.


A *(left) semimodule $M$ over a semiring $R$* is a commutative monoid $(M, \, +)$ together with a multiplication map $R \times M \to M$, denoted by $(r, \, m) \to rm$ and called *scalar multiplication*, which satisfy all axioms of a unitary ring except the axiom demanding the existence of additive inverses. Right semimodules are defined in a similar way.


For instance, the $\mathbb{N}$-semimodules are precisely the commutative monoids, exactly as the $\mathbb{Z}$-modules are the commutative groups. 


Another example is the half-space of points with non-negative coordinates in $\mathbb{R}^n$, that is in a natural way a $\mathbb{R}\_+$-semimodule.


The general theory of semimodules over semirings is discussed in the book *Semirings and their Applications* by Jonathan S. Golan, see [this googlebooks](https://books.google.it/books?id=DNdTF_4PzpoC&pg=PA149&lpg=PA149&dq=On%20modules%20over%20semirings&source=bl&ots=NJfM-yTTts&sig=oj2aDOJNTGY1JeGhWPn_mouUJog&hl=it&sa=X&ei=I700VeiuMsuRsgGgyIDYDQ&ved=0CGsQ6AEwCA#v=onepage&q=On%20modules%20over%20semirings&f=false) link.


In that book there is also the following nice example showing how of this construction appears when studying signal processing, see Example 14.5 p. 151.


Take the *tropical semiring* $R = (\mathbb{R} \cup \{\infty \}, \, \textrm{min}, \, +)$ and let $M = R^{\mathbb{R}}$, seen as a left $R$-semimodule. Then the elements of $M$ are the *signals*, the addition in $M$ corresponds to parallel composition of signals and the scalar multiplication gives the amplification of signals.


