Binary Operations with Axiomatic Systems

Question

Give an example for a set $G$ and a binary operation which does these:

1. The set is closed under the binary operation.
2. The binary operation is associative.
3. The binary operation has a neutral element.
4. There exists at least one element that does not have an inverse.

In other words, this is almost a group except the axiom about existence of inverses is negated.

I have really tried thinking about it for a long time (apx. 5 hours) and haven't got any idea yet.

Answer 1

Suppose such a set (say $S$) and binary operation (say $*$) exists. Let’s derive a contradiction.

We are assuming that we have an identity, say $e.$ So for any $a \in S$ we have that $a * e = a = e * a.$

But if the identity exists, then $e * e = e = e * e,$ and therefore $e$ is its own inverse. Which is a contradiction, because we are assuming that no element has inverse.

Therefore such set and binary operation can’t exist. $\square$

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Although, if you are looking for a set and a binary operation that satisfies axioms $1$, $2$ and $3$ but not all elements have an inverse (i.e., there is some element with no inverse), then consider the following.

Let $G$ be the set $G = \{0,1\}$ and consider as the binary operation, the usual multiplication.

Then it is straightforward to check that axioms $1$, $2$ and $3$ hold and not every element has an inverse (consider the element $0$ that doesn’t have an inverse under multiplication).

Answer 2

A very good answer to your question is to consider any field with multiplication operation. This operation is required to satisfty the first three axioms because that comes from the definition of a field. However, since there is no field with exactly one element, every field has a zero element which can not have a multiplicative inverse.