… an attempt to consolidate and synthesize all my previous work.

Leading question: why is it so natural in algebra to treat elements as both “objects” and “actions”? In $x-y$, have $x$ is the “object” and $y$ is the “action” (via $-$), and yet both are “elements” of $\mathbb R$. What gives?

In other words. Consider each of: (1) linear transformations and matrices; (2) the number line and translations, dilations; (3) the complex plane and rotations, dilations, translations. Each sounds like a set of objects paired with a set of actions. But, in fact, we express them as a set of objects paired with operations. Why?

• Usually think of operations as functions $S^n \to S$, but equivalent is to think of them as functions $S \to (S^{n-1} \to S)$.

  • Define notation $(\star \ a) = (x \mapsto x \star a)$ and likewise for $(a\ \star)$
  • Natural in first-style: $(\Sigma^\star, \oplus)$ (strings under concat), $(A \to A, \circ)$, $(\text{Set}, \cup)$, $(\mathbb N, +)$, etc
  • Natural in second-style: $(\mathbb R, +)$ (translation), $(\mathbb R \setminus \{0\}, \cdot)$ (dilation), $(\mathbb C, \text{rot})$ with $\text{rot}(a, b) = \frac{a}{\lvert a \rvert}b$ and $(\mathbb C, \text{dil})$ with $\text{dil}(a, b) = \lvert a \rvert b$. I claim this is natural by taking the view that $x \cdot 8$ is best thought of as “dilate $x$ by $8$”; now we can refer to the action “dilate by $8$” directly as $(\cdot \ 8)$.

• This gives rise to rephrasing of well-known properties:

  • Commutativity: $ab = ba$ to $(a \ \star) = (\star \ a)$, ie the left- and right- actions are the same
  • Idempotency $a(ax)$ to $(a\ \star)^{n,\ n \geq 1} = (a\ \star)^1$
  • Identity $\exists e : ea = a e = a$ to $\exists e : (e \ \star) = (\star \ e) = \text{id}$
  • Invertibility $\forall a \exists a^{-1} : aa^{-1} = a^{-1}a = e$ to $\forall a \exists a^{-1} : (\star \ a) \circ (\star\ a^{-1}) = (\star\ a^{-1}) \circ (\star\ a) = \text{id}$
  • Associativity (!) $a(bc) = (ab)c$ to $(\star\ ab) = (\star\ a) \circ (\star\ b)$
  • I personally find these to read far better than their conventional counterparts. Especially associativity, which now reads as “applying $\star$ is the same thing as composing it”
  • nb. Some of these rephrasings make assumptions. Invertibility, for instance, assumes commutativity.
  • Oh another one! We can rephrase

• Interesting property: translations are decided by a single value. That is, given $a, b \in \mathbb R$ and witness $x_0$ to $(+\ a)(x_0) = (+\ b)(x_0)$ then we have $a = b$. True of dilations and other things as well; not true in general.

• Incomplete thought: the naive way to think about translations is probably to reify translations on $\mathbb R$ as the set of “arrows” or “deltas” $\text{Tl} = \mathbb R^2 / \sim$ where $(a, b) \sim (x, y)$ when $b - a = y - x$

  Write an element of this quotient set as $a \leadsto b$, and let $\star : \mathbb R \times \text{Tl} \to \mathbb R$ such that $x \star (x_0 \leadsto x_f) = x + (x_f - x_0)$, so eg $2 \star (4 \leadsto 5) = 3$

  Then at some point we recognize that $\text{Tl} \cong \mathbb R$ and define $x + t = x \star (0 \leadsto t)$

  Something like that?

• One potential way to bring light to the situation is to take a step back and construct an abstraction where our values and actions are distinct

  Say, 3-tuples $(S, A, \ell)$ with $\ell : A \to (S \to S)$ — $S$ is ‘state’, $A$ is ‘action’

  Then we have natural examples like $(X, X \to X, \text{id})$ and $(\Sigma^\star, \mathbb N, \text{repeat})$ and $(\text {Set}(X), X, \text{add})$

  Call such 3-tuples, idk, threes, and also say a three is:

  • with identity if $\text{id} \in \ell(A)$
  • composable if exists an operator $\star$ where $\ell(a \star b) = \ell(a) \circ \ell(b)$
  • reversible if for each $a \in A$ exists an $a^{-1}$ with $\ell(a^{-1}) \circ \ell(a) = \text{id}$
  • preventable if for each $a \in A$ exists an $a^{-1}$ with $\ell(a) \circ \ell(a^{-1}) = \text{id}$
  • homogeneous if $S = A$
  • with homogenous composition if also $\ell = \star$

  Then we have (I think)

  • homogenous threes bij. with magmas
  • … and with homogeneous composition bij. with semigroups
  • … and with identity bij. with monoids

  That is, a magma/semigroup/monoid can be seen as degenerate cases of threes; a semigroup “is” a structure where values are actions and actions can be composed.

  Probably could rephrase much of algebra this way!

Unrelated, but I want to hang onto it:

Previous work

A commutative magma is a set $A$ paired with a mapping $[\cdot] : A \to A^A$. For $a, b \in A$, we write $a[b]$ for $b$.

A commutative magma …

• Has identity if exists $e \in A$ so that $a[e] = a$ for all $a$