Axiom talk:Semigroup Axioms

I don't understand what is the meaning of '=' in this context. Is it supposed to be set equality or something else (for example an equivalence relation)? --EnnioVisco (talk) 10:41, 11 September 2020 (UTC)

Nothing subtle or philosophically sophisticated. It just means "equals", which in this context means "is the same thing as".

$\circ$ is a binary operator, operating on two elements of $S$, whatever they are.

Let $f: S \times S \to S$ be the mapping defined as:

$\forall a, b \in S: \map f {a, b} = a \circ b$

Then it is seen that "$a \circ b = x$" means that $a \circ b$ is equal to, that is, "is the same thing as", the element $x$ of $S$ such that $\map f {a, b} = x$.

Hence: $\paren {a \circ b} \circ c = a \circ \paren {b \circ c}$ means, in this context, that whenever you have $3$ elements $a, b, c$ of $S$, and you apply the $\circ$ operator first to $\tuple {a, b}$ and then apply it to $\tuple {x, c}$, where $x$ is the result of applying $\circ$ to $\tuple {a, b}$, you always get the same result as if you apply the $\circ$ operator to $\tuple {b, c}$, and then apply it to $\tuple {a, y}$ where $y$ is the result of applying $\circ$ to $\tuple {b, c}$.

Or is your question more philosophically deep and subtle, and I have completely misunderstood the thrust of what you are asking? --prime mover (talk) 11:41, 11 September 2020 (UTC)

It might seem quite philosophical, but it's rather practical: the equality in $\map f {a, b} = a \circ b$ is used to say "to what thing" the symbol $\map f {a, b}$ refers to, while in $a \circ b = x$ it states that the two things are "the same". For what you are saying I believe that we are talking about the classical set equality.

Only if $a \circ b$ and $x$ are both sets. Which they are if and only if the semigroup under discussion is embedded in a universe which has been constructed entirely using some form of axiomatic set theory / class theory, and there is no such direct presupposition at this stage of development of the theory.

Apart from that, I can see no "practical" implications for the meaning of "equals" to be put to the question.

The practicality of the problem is related to defining some "special kinds" of Semigroups on a set $S$ where the equivalence relation $\RR$ holds. One might be interested in defining the associativity in terms of that relation. Note that in that case one would have that $a \circ b \RR x$, although it is not necessarily the case that $a \circ b = x$. Would this still be a proper Semiring? --EnnioVisco (talk) 13:18, 11 September 2020 (UTC)

Show us where such semigroups are defined, and if you are able to, direct our attention to whatever literature may be out there that sheds light on such.

I can see how it may be possible to redefine all our fundamental abstract algebraic constructs under the assumption that where we have "equals" we may instead want to use "an arbitrary and still-to-be-defined equivalence relation" but whether that offers any new insights would have to be demonstrated. Besides, I fear we'd completely lose all credibility with whatever undergraduates or masters-level mathematicians who would take one look, and, being unable to make head or tail of what they may well consider to be abstruse and overcomplicated claptrap, would dismiss $\mathsf{Pr} \infty \mathsf{fWiki}$ as a site whose purpose is for a highly-educated minority to show off how clever they are. We all know this is of course not the case, but $\mathsf{Pr} \infty \mathsf{fWiki}$ is already held in extreme ridicule by a vast majority of the mathematical community, and I really would like to avoid our reputation to deteriorate any further, however justified that may be.

In short: please direct us to some resources which explain in detail what you're talking about. Everything here is found in undergraduate level (and sometimes considerably higher) text books. I have read widely (although I admit not necessarily all that deeply), and I have never encountered a treatment anything like what you are suggesting. --prime mover (talk) 13:54, 11 September 2020 (UTC)