Is $\left(K, – \right)$ a group, semigroup, or monoid

group-theorymonoidsemigroups

My professor asked us to determine whether $\left(K, – \right)$ for $K$ is a set of integers is a group, semigroup, or monoid. Using my limited knowledge in the group theory here's 4 condition that I need to check

  1. For $a, b \in K$, $a-b$ is a uniquely defined element of $K$
  2. It satisfies associativity
  3. It has an identity element $e$
  4. For each $a \in K, \exists a^{-1} \in K$ then $a – a^{-1} = e$ and $a^{-1} – a = e$

I found that $\left(K, – \right)$ only satisfy (1). My question is $\left(K, – \right)$ a group, semigroup, or monoid? or neither?

Thank you

Best Answer

You're right in that only condition $1$ is satisfied.

As for determining what structure it is, a table on Wikipedia gives a list of various structures and the necessary properties for each:

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Within these definitions, $(K,-)$ is a magma. It is not a group or semigroup or monoid, since we don't have associativity in particular, among the other axioms they require.

Now, interestingly, there is a sense in which you can say structures satisfy invertibility without an identity. Namely, as for quasigroups, $(S,\ast)$ is invertible if and only if

$$(\forall a,b \in S)(\exists ! \; x,y \in S)(a \ast x = b \text{ and } y \ast a = b)$$

In the case of $(S,\ast) = (K,-)$, then this happens to hold: $x = a-b$ and $y = b-a$, and uniqueness is obvious. Within this definition, then you could go so far as to claim $(K,-)$ is a quasigroup.

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