Ehresmann theory and partition monoids

This article concerns Ehresmann structures in the partition monoid P X . Since contains symmetric and dual inverse monoids on same base set , it naturally semilattices of idempotents both submonoids. We show that one these leads to an structure while other does not. explore some consequences this (structural/combinatorial representation theoretic), particular characterise largest left-, right- two-sided restriction The new results are contrasted with known concerning relation monoids, a number interesting dualities arise, stemming from traditional philosophies semigroups as models partial symmetries (Vagner Preston) or block (FitzGerald Leech): “surjections between subsets” for relations become “injections quotients” partitions. also consider related diagram including rook state several open problems.