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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.

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    $\begingroup$ I don't really see how we could just do that, since we still have additive inverses in the field. $\endgroup$
    – Tobias Kildetoft
    Apr 20, 2015 at 8:30
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    $\begingroup$ And I am saying that we just get it right back since we would still be working over a field. $\endgroup$
    – Tobias Kildetoft
    Apr 20, 2015 at 8:34
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    $\begingroup$ Any field contains $-1$, so you still have $(-1)x$ for all $x$ in your space, and this is necessarily the additive inverse for $x$, unless you drop distributivity. $\endgroup$
    – Francesco Polizzi
    Apr 20, 2015 at 8:35
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    $\begingroup$ For this reason, the set of scalars cannot be a field. In fact, "modules over semirings" are the objects you are looking for. $\endgroup$
    – Francesco Polizzi
    Apr 20, 2015 at 8:38
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    $\begingroup$ As I explained in my answer below, the $\mathbb{N}$-semimodules are the commutative monoids, exactly as the $\mathbb{Z}$-modules are the commutative groups. $\mathbb{R}_+$-semimodules can be of course also defined, since the construction is general. $\endgroup$
    – Francesco Polizzi
    Apr 20, 2015 at 9:00

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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 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.

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    $\begingroup$ These things really should be called modules over semirings. If you are over a ring, you will get a module anyway so it is clear that the "deficiency" (or semi) is in the ``ring'' not the module. $\endgroup$
    – Benjamin Steinberg
    Apr 20, 2015 at 14:33
  • $\begingroup$ I agree with you, I just followed the terminology used in Golan's book. However, writing "$\mathbb{N}$-module" still makes me uncomfortable :-) $\endgroup$
    – Francesco Polizzi
    Apr 21, 2015 at 7:55

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