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I am a newbie in this field but what difference does taking monoid or group in the following definition of dynamic system make?

A tuple \begin{equation} (T,M,\phi) \end{equation}

is called dynamic system, where $T$ is additively written monoid (time), $M$ is a phase space and $\phi$ is an evolution operator

\begin{equation} \phi = U\subseteq T\times M \rightarrow M \end{equation}

of the system.

I have found another stronger definiton in which $T$ is said to be additive group.

Does it matter? Is the addition necessarily commutative?

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    $\begingroup$ I am at odds with $T$ being a group. How do you turn back the clock? $\endgroup$
    – Nobody
    Jul 22, 2018 at 7:54

1 Answer 1

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The difference is whether the time $t$ can only forward or backward as well. A group has inverses, a monoid does not need them. In particular, $\mathbb R$ is a group, $[0, \infty)$ (with addition) is not. If your dynamical system is reversible, you might want to use $\mathbb R$, otherwise only $[0,\infty)$.

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