Jech's Set Theory notation

Question

On page 74 of Set Theory (Third Millennium Edition) in Lemma 7.2 the term '$\subset$-chain' is used.

(ii) If $\mathcal C$ is a $\subset$-chain of filters on $S$, then $\bigcup\mathcal C$ is a filter on $S$.

I couldn't find any definition of it. What does it mean exactly?

Answer 1

A chain in a partially ordered set is a totally ordered subset. For any set $X$, the $\subseteq$ relation forms a partial order on that set, so a $\subseteq$-chain in $X$ is a subset $Y\subseteq X$ such that $\subseteq$ is a total order on $Y.$ In other words, for any $x,y \in Y,$ either $x\subseteq y$ or $y\subseteq x.$

Answer 2

A binary relation $R$ on a set $A$ is some (any) subset of $A\times A,$ although we often write $xRy$ for $(x,y)\in R.$ An $R$-chain is some (any) $C\subset A$ such that $(tRt'\lor t'Rt)$ whenever $t,t'$ are unequal members of $C.$

Some authors might also assume that a chain $C$ is not $\emptyset.$

In your Q, the set $A$ is the set of all filters on $S,$ and $R$ is $\subset.$

If $C$ is a non-empty set of filters on $S$ such that $\forall t,t'\in C\,(t\subset t'\lor t'\subset t)$ then $\cup C$ is a filter on $S.$ (We do not need to specify $t\ne t'$ because $\subset$ is a reflexive relation: $t\subset t.$)

I am using the modern standard notation that $t\subset t'\iff t\subseteqq t'.$