I've read this notion in T. Matolcsi's Models In Mechanics. Google finds nothing about it. I think that this notion has some other common name and I am curious, what is it.
The definition of cili algebra in that book is the following (p. 196).
A cili algebra is a normed star algebra such that the projections of its completion form an orthomodular $\sigma$-lattice under partial ordering and orthocomplementation introduced in 1.8.
The referenced 1.8 definition is this:
Let A be a star algebra. We say that $p\in P(A)$ is less than or equal to $q\in P(A)$, $p\le q$, if $pq=qp=p$.
The meaning of $P(A)$ is described so:
Let $A$ be a star algebra. $x\in A$ is called a projection if $x^*=x^2=x$. The set of projections is denoted by $P(A)$.
So, what is the usual name of "cili algebra" in the mathematical literature? Or if it's just that, then why isn't there any trace of it on the Internet? Where can be found this notion beyond Matolcsi's book?
The relevant pages of the book are these:
(the whole book can be downloaded from this link in djvu format.)