According to definitions given by the reviewer and J. B. Fugate [Diss. Math. 149 (1977; Zbl 0354.54017)], a proper subcontinuum \(K\) of a continuum \(X\) is a terminal continuum of \(X\) provided whenever \(A\) and \(B\) are proper subcontinua of \(X\) with \(X=A\cup B\) and \(A\cap K\neq \emptyset\neq B\cap K\) then either \(X = A \cup K\) or \(X = B \cup K\). A proper subcontinuum \(K\) of a continuum \(X\) is an absolutely terminal continuum of \(X\) provided \(K\) is a terminal continuum of each subcontinuum \(L\) of \(X\) which properly contains \(K\).
In this paper the author establishes that confluent mappings preserve the concept of an absolutely terminal subcontinuum and quasi-monotone mappings preserve the concept of a terminal subcontinuum. An example is given to show that quasi-monotone mappings do not preserve absolute terminality. Several questions are posed including the question of whether this implication is true for hereditarily quasi-monotone mappings.