Ehresmann idea and partition monoids
Authors: James East, Robert D. Gray
Summary: This text considerations Ehresmann buildings within the partition monoid PX. Since PX incorporates the symmetric and twin symmetric inverse monoids on the identical base set X, it naturally incorporates the semilattices of idempotents of each submonoids. We present that considered one of these semilattices results in an Ehresmann construction on PX whereas the opposite doesn’t. We discover some penalties of this (structural/combinatorial and illustration theoretic), and particularly characterise the most important left-, right- and two-sided restriction submonoids. The brand new outcomes are contrasted with recognized outcomes regarding relation monoids, and plenty of attention-grabbing dualities come up, stemming from the normal philosophies of inverse semigroups as fashions of partial symmetries (Vagner and Preston) or block symmetries (FitzGerald and Leech): “surjections between subsets” for relations develop into “injections between quotients” for partitions. We additionally think about some associated diagram monoids, together with rook partition monoids, and state a number of open issues.