I recently came across divided power algebras here: <http://amathew.wordpress.com/2012/05/27/lazards-theorem-ii/> It interests me because the free divided power algebra on one variable $x$, where $x^{(i)}$ models $x^i/i!$, seems like a good way of handling exponential and exponential-type series. (Divided power algebras that aren't generated by a single variable are more complicated to axiomatize, so I will stick to the one-variable case).


Quick summary: A system of divided powers for an element $x$ in an associative unital ring $R$ associates to each nonnegative integer $n$ a ring element $x^{(n)}$ with $x^{(0)} = 1$ and $x^{(1)} = x$ and satisfying the following condition for all $i,j \ge 0$:


$$x^{(i)}x^{(j)} = \binom{i + j}{i} x^{(i + j)}$$


As we can see, if $R$ is an algebra over the rationals, the obvious (and unique!) choice is to set $x^{(i)} = x^i/i!$ for all $i$.


Suppose I am interested in finding some $x^{[i]}$ that models $x^i/i$ instead of $x^i/i!$, i.e., I am interested in the kind of terms that appear in the expansions of logarithmic and inverse trig series. My best guess for the right analogue to consider is this: $x^{[i]}$ is defined for all positive integers $i$ with $x^{[1]} = x$ and satisfy the following condition. For each $i,j > 0$, write $ij/(i + j)$ as a reduced fraction $r/s$. Then, we must have:


$$rx^{[i]}x^{[j]} = sx^{[i + j]}$$


I'd like to know whether this structure has been studied in the past, and/or what names have been given to the structure in question. Also, I'd be interested if anybody has opinions on whether the above is a reasonable way of trying to model $x^i/i$.



What's the name for the analogue of divided power algebras for x^i/i?
First of all, let me remark that one normally speaks of a divided power structure *on an ideal* $I\subset R$ (rather than on the whole $R$), by asking the existence of *divided powers* only for elements in the ideal; the requirement is then that all divided powers but the $0$-th also lie in the ideal.


I think that your divided power structure would not be very useful – let me call it logarithmic divided power as Liz suggested. If you have the "usual" one on your ideal $I$ (let me denote it by $\gamma$ as in Berthelot&Ogus' book) you can immediately deduce a logarithmic one (call it $\lambda$) by setting
$$
\lambda\_i(x)=(i-1)!\gamma\_i(x)
$$
for all $x\in I$. Since $(i-1)!\in\mathbb{Z}$, this defines a unique logarithmic divided power structure on $I$.


But unfortunately, your logarithmic powers are weaker than usual ones: for instance, we know (see Berthelot&Ogus' book, page 3.2, Lemma 3.3) that if $V$ is a discrete valuation ring of mixed characteristic $(0,p)$ and uniformizer $(\pi)$, we can put a divided power structure on $(\pi^k)$ if and only if the absolute ramification index $e$ of $V$ verifies $e\leq k(p-1)$ (only the case $k=1$ is treated in the book, but computations are analogous for $k\geq 1$). If you want to stress that your logarithmic structure exist, you only need to insist that $v\_\pi[\lambda\_i(\pi^k)]\geq 1$ for all $i\geq 1$. Computing explicitly, this corresponds to
$$
e\leq \frac{ik-1}{v\_p(i)}
$$
which, by monotonicity of $i/v\_p(i)$, is satisfied as soon as $e\leq kp-1$. Since this is (in general) bigger than $k(p-1)$ you see that there are rings admitting one structure but not the other; since in crystalline cohomology and related topics one really needs exponentials (so, dividing out by $n!$), yours are too weak to be useful.


You started your question saying "Suppose I am interested in finding...": are you really?


