Flatness properties of diagonal acts over monoids

Acts Over Semigroups

“…In Theorem 4.9 of [7], it is proved that for an infinite epigroup S bdr S > 1, which means that this inequality also holds for infinite periodic and infinite locally finite semigroups. In [18], the connections between these notions were studied for diagonal acts, and also the question of when a diagonal act is free, projective, ((principally) weakly) flat, etc. Theorem 8.13 ([18,Proposition 5]).…”

“…In [18], the connections between these notions were studied for diagonal acts, and also the question of when a diagonal act is free, projective, ((principally) weakly) flat, etc. Theorem 8.13 ([18,Proposition 5]). Let S be a monoid.…”

“…Let S be a monoid. The diagonal act (S × S) S is weakly flat if and only if it is principally weakly flat, and for any a, b ∈ S, there exists c ∈ S such that Sa ∩ Sb ⊆ (Sa ∩ Sb)c. Theorem 8.14 ([18,Theorem 6]). Let S be the semigroup of injective maps α : N → N such that |N\Nα| = |N| with multiplication from left to right.…”

On direct products of S-posets satisfying flatness properties

“…Meanwhile, Bulman-Fleming and McDowell [4] defined a monoid to be right coherent if every direct product of flat S -acts is flat, and they obtained some results when S Γ is (principally) weakly flat for a monoid S . In [2], Bulman-Fleming and Gilmour discussed when S × S has certain flatness properties. Then in [9], principally weakly left coherent monoids were characterized as monoids over which direct products of nonempty families of principally weakly flat right S-acts are principally weakly flat.…”