Comments on Would sesquation have identity element 0, 1, or something else?
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Would sesquation have identity element 0, 1, or something else?
If we allow a schematic of hyperoperations $\uparrow^a$ to take values for $a \not\in \mathbb{N}$, then setting $a = 0 := Succ(n)$, sesquation occurs for $\uparrow^{\frac{3}{2}}$. This is intended to be, as literally as can be, intermediary between $+$ and $\times$.
Now what would its identity element be, if it has one? Would it be $0$, like with addition, or $1$, like with multiplication? Or would it somehow be $\frac{1}{2}$? Would it "oscillate" between $0$ and $1$, or be "superpositioned" in both states? Might sesquation not even admit of an identity element? The nLab entry on identity elements reads at one point:
... a unit law is the statement that a given operation has an identity element.
... so I don't know that I should automatically guess that sesquation "should have" an identity element anyway?
Motivation: I'm experimenting with a form of "passing to the limit" for $\frac{2xh + h^2}{h}$ where $+$ is replaced by a variable hyperoperator $\uparrow^q$ such that $h$ covaries with $q$ so that $h \rightarrow 0$ when $q \rightarrow 1$ ("treat $h$ like $0$ when adding it to $2x$") and then $h \rightarrow 1$ when $q \rightarrow 2$ ("treat $h$ like $1$ when multiplying it by $2x$"). The "reason for this" is that when $x = 2$, the formula for the slope can be "toggled between" $2x, 2 + x$. (I don't have much of anything like a very clean explanation for why "toggling between" addition and multiplication should be generalized over the formula in its consolidated form; for now, my intuition is just screaming at me that we can simulate the old practice of "neglecting" $h$ by this means.)
But what would this "passing to the limit" for $q$ be? Would we superpose the values $1, 2$ for $q$ or would we "pass through" hyperoperations purportedly in-between $\uparrow^1$ and $\uparrow^2$? In the latter event, I want to have a more definite sense for sesquation, esp. in terms of whether it has a unique identity element or not. (I'm imagining an "abstract algebraic" or "category-theoretic" argument where $h$ is "able to be neglected" because it is a term grounded in identity elements, "being $0$" relative to when $0$ is an identity element and "being non-$0$" when e.g. $1$ is an identity element.)
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