Relationships Among Some Concepts of Bivariate Dependence

We consider some unresolved relationships among various notions of bivariate dependence. In particular we show that $Plbrack T > t mid S > s brack uparrow$ in $s$ (or alternately, $Plbrack T leqq t mid S leqq s brack downarrow$ in $s$) implies $S, T$ are associated, i.e. $operatorname{Cov} lbrack f(S, T), g(S, T) brack geqq 0$ for all non-decreasing $f$ and $g$.