If $(M,+)$ is a monoid then it obeys the laws:

$$m_1 + 0 = 0 + m_1 = m_1$$ $$m_1+(m_2+m_3)=(m_1+m_2)+m_3$$

But what if I have a structure $(A,+)$ that is almost a monoid, but not quite. For example, maybe it has an associated norm and obeys the laws:

$$|a + 0| < \delta$$ $$|0 + a| < \delta$$ $$|a_1+(a_2+a_3)| - |(a_1+a_2)+a_3| < \delta$$

Thus, we don't have an identity or associativity, but we have things that act almost like identities and associativity. There are probably many other types of "almost monoids" (for example by multiplicative error instead of additive error). One simple example of a structure with these types of properties is floating point numbers on computers (although they probably need some more laws to handle infinities).

I'm working with some structures that have similar properties to this, and I'm wondering if these types of approximations have been studied before at all? I'm currently focusing on just the monoid's associativity property, but I'd be interested in other related approximate structures as well for ideas.