We consider the posets of equivalence relations on finite sets under the
standard embedding ordering and under the consecutive embedding ordering. In
the latter case, the relations are also assumed to have an underlying linear
order, which governs consecutive embeddings. For each poset we ask the well
quasi-order and atomicity decidability questions: Given finitely many
equivalence relations Ο1β,β¦,Οkβ, is the downward closed set
Av(Ο1β,β¦,Οkβ) consisting of all equivalence relations which do not
contain any of Ο1β,β¦,Οkβ: (a) well-quasi-ordered, meaning that it
contains no infinite antichains? and (b) atomic, meaning that it is not a union
of two proper downward closed subsets, or, equivalently, that it satisfies the
joint embedding property