On a relation between $λ$-full well-ordered sets and weakly compact cardinals

Abstract

We prove, via transfinite recursion, the existence, inside any linearly ordered set of appropriate regular cardinality $λ$, of a particular kind of well-ordered subsets characterized by the property of $λ$-fullness. Let $H$ be a set of regular cardinals: by using our results about well-ordered $λ$-full sets we show that if $\inf H$ is a weakly compact cardinal, then, for every LOTS $X$, $H$-compactness is equivalent to the nonexistence of gaps of types in $H$.