# Reasoning about “approximately” associative structures and “almost monoids”.

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.

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If the space has a norm or valuation, you might consider arbitrary operations f which have the properties you desire (being associative or commutative) and such that f (or some term involving f) is within delta of your desired term. Then you can ask how near your algebra is to, say, a commutative monoid. Or you can ask how far f is from the set of terms of appropriate arity of your algebra. For the latter, Kolmogorov and Arnol'd solved a Hilbert problem about representing continuous functions just using addition and unary functions. Gerhard "Ask Me About System Design" Paseman, 2013.05.30 – Gerhard Paseman May 30 '13 at 20:20
It is barely possible that something like this was covered in topological general algebras, studied by Walter Taylor. I don't have any concrete suggestions for you at this point; just vague ideas which may be unhelpful. If you had more of a goal (e.g. normed clone theory), I might suggest a search term for you. Gerhard "Ask Me About Wild Ideas" Paseman, 2013.05.30 – Gerhard Paseman May 30 '13 at 20:25