Dimension and universality on frames. (Q5963959)

The author continues his study (sometimes jointly with other authors) of universality of frames. In this paper he introduces the notion of a saturated class of frames. The definition is too elaborate to include in this review. Suffice to say the class of all frames of a given weight is saturated; and, for a given cardinal \(\tau\), intersection of not more than \(\tau\) many saturated classes of frames of weight not more than \(\tau\) is also a saturated class. The latter is not a trivial observation, but a theorem that requires some muscle to prove. Another somewhat unexpected result is that in any saturated class of frames there are universal elements.NEWLINENEWLINE In the second part of the paper the author associates with each frame \(L\) a cardinal, denoted \(\text{dec}(L)\), which he calls the decomposition invariant of \(L\). It is a dimension-like invariant. Indeed, he shows, for instance, that \(\text{dec}(L)=0\) precisely when \(L\) is 0-dimensional, in the sense that \(L\) is join-generated by its complemented elements. One of the main theorems is that, for fixed cardinals \(\mu\leq\tau\), the class of all frames \(L\) of weight \(\leq\tau\) with \(\text{dec}(L)\leq\mu\) is a saturated class; which means that in this class there are universal elements.