In Jónsson and Tarski’s (1951) paper Boolean Algebras with Operators, Part I from the American Journal of Mathematics, they write formulae such as
$L_i = underset{u}{mathbf{E}} , [u in At^m text{ and } u leq x^{(i)}]$
and
$K = underset{u}{mathbf{E}} , [y geq u in At^m]$,
without explaining this $underset{u}{mathbf{E}}$ notation. From the context, I guess these define sets, i.e., they respectively mean
$L_i = {u mid u in At^m text{ and } u leq x^{(i)} }$
and
$K = {u mid y geq u in At^m }$.
Am I correct in this?
Also, is this notation something common that mathematicians generally understand? What does E stand for, and where did this notation originate? Are there good books/articles/webpages where I can learn about this notation?
I would greatly appreciate your help!
Additional note. To give some context, on p. 900, following the first formula, they proceed to define $K = bigcup_{i in I} L_i$ (which has nothing to do with the $K$ in the second of the formulae above) and say that $u in K$ if and only if $sum_{i in I} x^{(i)} geq u in At^m$. In order for this equivalence to hold, it seems to me that $L_i = {u mid u in At^m text{ and } u leq x^{(i)} }$.
$At$ denotes the set consisting of $0$ and all the atoms of a Boolean algebra $A$. For my interpretation to make sense, however, I suppose that $At^m$ should be the set consisting of $0$ and all the atoms of $A^m$, and not the $m$-times product of $At$.