Semigroup where the Binary Operation is not Associative.

Question

I am working on my functional composition, which has the associative property, to show if a given pair is a semigroup or not. I believe all Semigroups have to have a binary operation that is associative.

Am I correct?

Answer 1

In mathematics, a semigroup is an algebraic structure consisting of a set together with an associative binary operation

-Wikipedia, 2015

If you're interested in a set with a not-necessarily-associative binary operation, you're looking for a "magma".

Answer 2

Associativity is part of the definition of semigroup, so yes, all semigroups are associative.

If you need to show that some set with an operation on it is a semigroup, though, you can't take it for granted. You must instead show that the operation is defined for every pair of elements and that it is indeed associative.