On monoids over which all generators satisfy a flatness property

On classification of pomonoids by properties of generators

“…Assertions (i), (ii) and (iii) are equivalent by Theorem 1.1 and Lemma 2.6. Implications (iii) ⇒ (iv) and (iv) ⇒ (v) is similar to the act case (see [15]). We remark that left po-cancellative pomonoids are examples of pomonoids over which all generators satisfy Condition (E).…”

“…Assertions (i), (ii) and (iii) are equivalent by Theorem 1.1 and Lemma 5.1. Implications (i) ⇒ (iv) and (iv) ⇒ (i) is similar to the act case (see [15]).…”

“…Thena ⊗ s ≤ a ⊗ sxs ≤ as ⊗ xs ≤ a ′ s ⊗ xs ≤ a ′ ⊗ sxs ≤ a ′ ⊗ s in A ⊗ S Ss.This proves that A s is principally weakly po-flat by[19, Proposition 2.7]. Sy, such that p ≤ q, ax ≤ a ′′ p and a ′′ q ≤ a ′ y.Similar to the definition in S-acts (see[15, Definition 3.2]), we have Definition 4.2. A right S-poset A S is called almost weakly po-flat if A S is principally weakly po-flat and satisfies condition (W ′ ) if ax ≤ a ′ y and Sx ∩ (Sy] = ∅ for a, a ′ ∈ A S , x, y ∈ S, then there exist a ′′ ∈ A S , p ∈ Sx and q ∈ Sy such that p ≤ q, ax ≤ a ′′ p and a ′′ q ≤ a ′ y.…”

“…A right congruence ρ on S S is called a right subannihilator congruence if ρ ≤ ker λ s (the left translation by s) for some s ∈ S (see also [15] for unordered case). It is easy to check that a right congruence ρ on S S is a right subannihilator congruence if and only if Pos S (S S /ρ, S S ) = ∅.…”

“…Kilp and Knauer, in [10], investigated monoids over which all generators in the category of right S-acts satisfy a special flatness property. Continuing this study, Sedaghatjoo, in [15], investigated the corresponding problem for weak flatness, Condition (E), and regularity in the category of right S-acts. The purpose of the present article is to discuss pomonoids over which all generators in the category Pos-S satisfy a special property, that is related to flatness like Condition (E), weak po-flatness, principal weak po-flatness, po-torsion freeness and I-regularity.…”