# Monoid in general dynamic system definition

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 $$(T,M,\phi)$$

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

$$\phi = U\subseteq T\times M \rightarrow M$$

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?

• I am at odds with $T$ being a group. How do you turn back the clock? Jul 22, 2018 at 7:54

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 $[0,\infty)$.