INDECOMPOSABLE, PROJECTIVE, AND FLATS-POSETS

On Condition $(PWP)_{w}$ for $S$-posets

“…The tensor product A⊗ S B of a right S -poset A S and a left S -poset S B is a poset that can be constructed in a standard way (see [13] for details) so that the map A × B → A ⊗ S B sending (a, b) to a ⊗ b is balanced, monotonic in both variables, and universal among balanced, monotonic maps from A × B into posets. The order relation on A ⊗ S B can be described as follows: .…”

Ordered left PP monoids

“…We note that if an S-poset P is projective, then P ∼ = i∈I Se i where e 2 i = e i ∈ S, i ∈ I (see [19]), thus P is semifree. From Corollary 2.7, we have the following theorem.…”

“…An S-poset P is called projective if for any S-surmorphism π : A → B and any S-morphism ϕ : P → B there exists an S-morphism ψ : P → A such that ϕ = πψ. In [19], we proved that an S-poset P is projective if and only if P ∼ = i∈I Se i where e 2 i = e i ∈ S, i ∈ I. A free S-poset is the coproduct i∈I S i , where each S i is S-isomorphic to the S-poset S, a semifree S-poset is a coproduct of cyclic S-posets.…”

“…Partially ordered acts over a pomonoid S, or S-posets, appear naturally in the study of mappings between posets. In [19], projective S-posets are discussed and ordered lpp monoids are introduced. We can see the class of ordered lpp monoids contains all ordered right cancellative monoids and all regular pomonoids.…”

On direct products of S-posets satisfying flatness properties

“…More information about tensor products in S-posets can be found in [12]. A right S -poset A S is weakly po-flat if a ⊗ s ≤ a ′ ⊗ t in A S ⊗ S (equivalently, as ≤ a ′ t ) implies that the same inequality holds also in A S ⊗ S (Ss ∪ St) for a, a ′ ∈ A S , s, t ∈ S .…”