It is well-known that monoids are stunningly ubiquitous in programing. They are so ubiquitous and so useful that I, as a 'hobby project', am working on a system that is completely based on their properties (distributed data aggregation). To make the system useful I need useful monoids :)

I already know of these:

• Numeric or matrix sum
• Numeric or matrix product
• Minimum or maximum under a total order with a top or bottom element (more generally, join or meet in a bounded lattice, or even more generally, product or coproduct in a category)
• Set union
• Map union where conflicting values are joined using a monoid
• Intersection of subsets of a finite set (or just set intersection if we speak about semigroups)
• Intersection of maps with a bounded key domain (same here)
• Merge of sorted sequences, perhaps with joining key-equal values in a different monoid/semigroup
• Bounded merge of sorted lists (same as above, but we take the top N of the result)
• Cartesian product of two monoids or semigroups
• List concatenation
• Endomorphism composition.

Now, let us define a quasi-property of an operation as a property that holds up to an equivalence relation. For example, list concatenation is quasi-commutative if we consider lists of equal length or with identical contents up to permutation to be equivalent.

Here are some quasi-monoids and quasi-commutative monoids and semigroups:

• Any (a+b = a or b, if we consider all elements of the carrier set to be equivalent)
• Any satisfying predicate (a+b = the one of a and b that is non-null and satisfies some predicate P, if none does then null; if we consider all elements satisfying P equivalent)
• Bounded mixture of random samples (xs+ys = a random sample of size N from the concatenation of xs and ys; if we consider any two samples with the same distribution as the whole dataset to be equivalent)
• Bounded mixture of weighted random samples

Which others do exist?