Abstract. A mapping $\alpha $ on a poset $P$ is {\it decreasing} if $p \alpha\leq p$ for $p \in P$. For a class of finite posets [lattices] $\cal U$ let ${\rm DEnd} \cal U$ stand for the class of the semigroups which can be faithfully represented by decreasing order [lattice] endomorphisms of posets [lattices] from $\cal U$. We consider the classes ${\rm DEnd} \cal U$ for various $\cal U$ in comparison with the two classes ${\rm DEnd} {\cal C}$ and ${\rm DEnd} {\cal P}$ studied earlier, where ${\cal C}$ [${\cal P}$] stands for the class of all finite chains [posets].

AMS Subject Classification (1991): 20M20, 20M30, 06A07

Received April 7, 1998 and in revised form September 28, 1998. (Registered under 2673/2009.)