Forbidden configurations for coherency

Right (and left) coherency and right (and left) weak coherency are natural finitary conditions for monoids. Determining whether or not a given monoid has any of these properties is historically a difficult problem. This paper has several aims, centering around the well-studied class of right (and dually left) $E$-Ehresmann monoids, being one of the broadest classes of monoids containing a semilattice of idempotents. First, we exhibit a particular configuration of elements in a monoid subsemigroup of a right (respectively, left) $E$-Ehresmann monoid, relative to the Ehresmann structure of the overmonoid, that prohibits left (respectively, right) coherence. Second, we apply this technique in a number of different situations. We show that the free Ehresmann monoid of rank at least $2$ is neither left nor right coherent, and that the free left Ehresmann monoid is not left coherent. We demonstrate the utility of our technique in the case where the overmonoid is an $E$-unitary inverse monoid, and apply this to both new situations and to recover the existing results. Namely, free inverse monoids and free ample monoids of rank at least 2 are neither left nor right coherent, and free left ample monoids of rank at least $2$ is are not left coherent. Next, in a positive direction, we demonstrate that every free left Ehresmann monoid is weakly coherent. Our final result is of a different nature. Ehresmann monoids form a variety of monoids with an enriched signature. Viewed as a bi-unary monoid (respectively, unary monoid), a free Ehresmann monoid (respectively, free left Ehresmann monoid) does not embed into an inverse monoid. We show that viewed as a monoid (the standpoint of this paper) every free Ehresmann monoid (and hence also every free left Ehresmann monoid) embeds into an $E$-unitary inverse monoid.