On regular torsionless S-posets
This paper shall be concerned with the notion of regular torsionless in the category of S-posets. Besides elementary basic properties of regular torsionless S-posets, we consider cyclic regular torsionless S-posets and also study when regular torsionless property is preserved under coproducts. Then...
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irk-123456789-1883752023-02-27T01:27:49Z On regular torsionless S-posets Khosravi, R. This paper shall be concerned with the notion of regular torsionless in the category of S-posets. Besides elementary basic properties of regular torsionless S-posets, we consider cyclic regular torsionless S-posets and also study when regular torsionless property is preserved under coproducts. Then we characterize pomonoids over which all free or projective S-posets are regular torsionless. Finally, we present conditions on S which follow if all regular torsionless S-posets are principally weakly po-flat, weakly po-flat, strongly flat, or projective. 2018 Article On regular torsionless S-posets / R. Khosravi // Algebra and Discrete Mathematics. — 2018. — Vol. 26, № 1. — С. 76–89. — Бібліогр.: 11 назв. — англ. 1726-3255 2010 MSC: Primary 06F05; Secondary 20M30. http://dspace.nbuv.gov.ua/handle/123456789/188375 en Algebra and Discrete Mathematics Інститут прикладної математики і механіки НАН України |
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This paper shall be concerned with the notion of regular torsionless in the category of S-posets. Besides elementary basic properties of regular torsionless S-posets, we consider cyclic regular torsionless S-posets and also study when regular torsionless property is preserved under coproducts. Then we characterize pomonoids over which all free or projective S-posets are regular torsionless. Finally, we present conditions on S which follow if all regular torsionless S-posets are principally weakly po-flat, weakly po-flat, strongly flat, or projective. |
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Khosravi, R. On regular torsionless S-posets Algebra and Discrete Mathematics |
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Khosravi, R. |
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Khosravi, R. |
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On regular torsionless S-posets |
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On regular torsionless S-posets |
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On regular torsionless S-posets |
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On regular torsionless S-posets |
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On regular torsionless S-posets |
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on regular torsionless s-posets |
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Інститут прикладної математики і механіки НАН України |
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2018 |
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On regular torsionless S-posets / R. Khosravi // Algebra and Discrete Mathematics. — 2018. — Vol. 26, № 1. — С. 76–89. — Бібліогр.: 11 назв. — англ. |
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Algebra and Discrete Mathematics |
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AT khosravir onregulartorsionlesssposets |
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2025-07-16T10:24:18Z |
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2025-07-16T10:24:18Z |
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1837798742486417408 |
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“adm-n3” — 2018/10/20 — 9:02 — page 76 — #82
Algebra and Discrete Mathematics RESEARCH ARTICLE
Volume 26 (2018). Number 1, pp. 76–89
c© Journal “Algebra and Discrete Mathematics”
On regular torsionless S-posets
Roghaieh Khosravi
Communicated by V. Lyubashenko
Abstract. This paper shall be concerned with the notion of
regular torsionless in the category of S-posets. Besides elementary
basic properties of regular torsionless S-posets, we consider cyclic
regular torsionless S-posets and also study when regular torsion-
less property is preserved under coproducts. Then we characterize
pomonoids over which all free or projective S-posets are regular
torsionless. Finally, we present conditions on S which follow if all
regular torsionless S-posets are principally weakly po-flat, weakly
po-flat, strongly flat, or projective.
Introduction
Over the past three decades, an extensive theory of the properties
of S-acts has been developed. A comprehensive survey of this area was
published in 2000 by Kilp et al. in [3]. The category of S-posets, as
the ordered version of the category of S-acts, recently has captured the
interest of some mathematicians [1,2]. There are many papers attempting
to generalize some properties including projectivity and various kinds of
flatness properties from S-acts to S-posets (see, for example, [9–11]). In
the category of S-acts, torsionless right acts over a monoid S are acts AS
such that the natural homomorphism ϕA from AS into its second dual is
injective. As far as we know torsionless acts are introduced and considered
in [4]. In this paper we introduce (regular) torsionless S-posets as the
2010 MSC: Primary 06F05; Secondary 20M30.
Key words and phrases: S-posets, pomonoids, regular torsionless, projective,
flat.
“adm-n3” — 2018/10/20 — 9:02 — page 77 — #83
R. Khosravi 77
S-posets for which ϕA is a (regular) monomorphism. In Section 1, we give
some basic properties of regular torsionless S-posets. In particular, we
discuss when regular torsionless property is preserved under coproducts
and give the conditions over which the amalgamated coproduct A(I), cyclic
S-posets and Rees factor S-posets are regular torsionless. In Section 2,
pomonoids over which free or projective S-posets are regular torsionless
are characterized. Finally, some necessary conditions for regular torsionless
S-posets to satisfy some flatness properties are given.
First we give some preliminaries needed in the sequel. A monoid S
endowed with a partial order, compatible with the binary operation, is
called a pomonoid. For a pomonoid S a right S-poset is a poset A which
is also a right S-act whose action is monotone in both arguments. A
right S-subposet of a right S-poset AS is a nonempty subset of A that
is closed under the action of S. Moreover, S-poset morphisms or simply
S-morphisms are monotone maps between S-posets which preserve actions.
The class of right S-posets and S-morphisms form a category, denoted by
POS-S. Let S be a pomonoid and I a nonempty set of S. In the following
statements, if we say I is a right ideal of S, it only means IS ⊆ S. A right
poideal of a pomonoid S is a subset I of S which is both a right ideal
(IS ⊆ I) and a poset ideal (that is, a 6 b, b ∈ I imply a ∈ I). For s ∈ S,
(sS] is called a principal right poideal of S.
Let A be a right S-poset. An S-poset congruence θ on A is a right
S-act congruence with the property that the S-act A/θ can be made into
an S-poset in such a way that the natural map A → A/θ is an S-morphism.
For an S-act congruence θ on A we write a 6θ a
′ if the so-called θ-chain
a 6 a1θb1 6 a2θb2 . . . 6 anθbn 6 a′
from a to a′ exists in A, where ai, bi ∈ A, 1 6 i 6 n. It can be shown
that an S-act congruence θ on a right S-poset A is an S-poset congruence
if and only if aθa′ whenever a 6θ a′ 6θ a. For two S-poset congruences
θ, σ on AS , we say that θ 6 σ if x 6θ y implies x 6σ y for each x, y ∈ AS .
Let H ⊆ A × A. Then a 6α(H) b if and only if a 6 b or there exist
n > 1, (ci, di) ∈ H, si ∈ S, 1 6 i 6 n such that
a 6 c1s1 d1s1 6 c2s2 . . . dnsn 6 b.
The relation ν(H) given by a ν(H) b if and only if a 6α(H) b 6α(H) a is
the S-poset congruence induced by H.
The subkernel of an S-poset morphism f : AS → BS is defined by
−→
kerf := {(a, a′) ∈ A × A : f(a) 6 f(a′)}. Then ν(
−→
kerf) = ker f :=
“adm-n3” — 2018/10/20 — 9:02 — page 78 — #84
78 On regular torsionless S-posets
{(a, a′) ∈ A × A : f(a) = f(a′)}, and in AS/ker f, [a]ker f 6 [a′]ker f
if and only if f(a) 6 f(a′). An S-morphism f : A → B is a regular
monomorphism if it is an order-embedding, i.e., a 6 a′ ⇐⇒ f(a) 6 f(a′),
for all a, a′ ∈ A. Obviously, f is a regular monomorphism if and only
if
−→
kerf = ξA = {(a, a′) ∈ A × A| a 6 a′}. S-isomorphism means both
monomorphism and epimorphism. A surjective order embedding of S-
posets is called an order isomorphism.
Now we recall the concepts we use in Section 2, more information on
these concepts can be found in [1,10]. A right S-poset AS is weakly po-flat
if a⊗ s 6 a′ ⊗ t in AS ⊗ S implies that the same inequality holds also in
AS ⊗ S(Ss ∪ St) for a, a′ ∈ AS , s, t ∈ S. A right S-poset AS is principally
weakly po-flat if as 6 a′s implies that a ⊗ s 6 a′ ⊗ s in AS ⊗ SSs for
a, a′ ∈ AS , s ∈ S. A right S-poset AS satisfies the condition
(P) if, for all a, b ∈ A and s, t ∈ S, as 6 bt implies a = a′u, b = a′v for
some a′ ∈ A, u, v ∈ S with us 6 vt,
and it satisfies condition
(E) if, for all a ∈ A and s, t ∈ S, as 6 at implies a = a′u for some
a′ ∈ A, u ∈ S with us 6 ut.
A right S-poset is called strongly flat if it satisfies both conditions (P)
and (E). Projectivity and freeness are defined in the standard categorical
manner.
We recall from [11] that an S-subposet B of an S-poset A is called
strongly convex if for any a ∈ A and any b ∈ B, a 6 b implies a ∈ B.
By [11, Theorem 2.3], every S-poset A is uniquely decomposable into
a disjoint union of strongly convex indecomposable S-subposets. Now,
using [11, Theorem 3.4], for a projective S-poset P all its strongly convex
indecomposable S-subposets are cyclic projective. Moreover, by [11, Propo-
sition 3.2], aS is projective if and only if there exists an idempotent element
e ∈ S such that a = ae, and as 6 at implies es 6 et for each s, t ∈ S, in
this case a is called left e-po-cancellable.
Concluding this section we give brief results describing when direct
products of S are projective, which will be needed in Section 2. If S is a
pomonoid, the cartesian product SI is a right S-poset equipped with the
order and the action componentwise where I is a non-empty set. Moreover,
(si)i∈I ∈ SI is denoted simply by (si)I or ~s, and the right S-poset S × S
will be denoted by D(S).
We say that two elements a, b of an S-poset AS are comparable if
a 6 b or b 6 a and denote this relation by a ∦ b.
“adm-n3” — 2018/10/20 — 9:02 — page 79 — #85
R. Khosravi 79
Definition 1. Let S be a pomonoid and I be a nonempty set. For each
~a ∈ SI we define:
L(~a) = {~b ∈ SI | biai ∦ bjaj , ∀i, j ∈ I}.
Note that L(~a) is either empty or left S-subposets of SI .
Proposition 1. Let S be a pomonoid such that for each J 6= ∅, SJ is
projective right S-poset. Then for every nonempty set I and every ~a ∈ SI ,
L(~a) is either empty or cyclic left S-poset of SI .
Proof. Suppose that L(~a) 6= ∅ for ~a ∈ SI . Put L(~a) = {~bj | j ∈ J}, where
~bj = (bi
j)I for each j ∈ J . Consider (bi
j)J and (bk
j)J for i, k ∈ I. We have
(bi
j)J ∦ 1(bi
j)J , (bi
j)Jai ∦ 1(bk
j)Jak and (bk
j)J ∦ 1(bk
j)J .
So by [11, Proposition 2.6], these elements belong to a strongly convex
indecomposable S-subposet of SJ . As we mentioned earlier, using Theorem
3.4 and Proposition 3.2 of [11] since SJ is projective, let this indecompos-
able S-subposet be ~pS, where ~p = (pj)J , and ~p be left e-po-cancellable for
some idempotent e ∈ S. Then for each i ∈ I there exists ui ∈ S such that
(bi
j)J = ~pui. Now, we get
~puiai = (bi
j)Jai ∦ 1(bk
j)Jak = ~pukak,
which implies that euiai ∦ eukak. Thus, ~q = (eui)I ∈ L(~a). Moreover, for
~bj ∈ L(~a), we have
~bj = (bi
j)I = pjui = pjeui = pj(eui)I = pj~q.
Therefore, L(~a) = S~q is cyclic.
Corollary 1. Let S be a pomonoid such that for each J 6= ∅, SJ is
projective right S-poset. Then S is right po-cancellative.
Proof. Let a, s, t ∈ S be such that sa 6 ta and s 6= t. Take ~a = (a)S ∈ SS .
Then {s, t}S ⊆ L(~a), which gives that |L(~a)| > 2|S|. Thus L(~a) is not
cyclic, which is a contradiction to the previous proposition.
“adm-n3” — 2018/10/20 — 9:02 — page 80 — #86
80 On regular torsionless S-posets
1. Regular torsionless S-posets
In this section we introduce regular torsionless S-posets and give some
basic results. First we need some preliminaries.
Let AS be a right S-poset. Then hom(AS , SS), if not empty, is a
left S-poset under the left multiplication (sf)(a) = sf(a) and the order
f 6 g ⇐⇒ (∀a ∈ AS)(f(a) 6 g(a)) for every f, g ∈ hom(AS , SS), s ∈ S
and a ∈ AS . The left S-poset hom(AS , SS) is called the dual of AS and
denoted by (AS)
∗. Note that hom(AS , SS) can be considered as an S-
subposet of (SS)
I where I has the same cardinality as the underlying
set of AS . Moreover, hom(−S , SS) is a contravariant functor from the
category of all right S-posets POS-S into the category all left S-posets
S-POS. This functor is called the dual functor. If f : BS → AS is an
S-morphism, then hom(f, SS) will be denoted by f∗. Right-left dually, if
SA is a left S-poset then hom(SA, SS) is a right S-poset.
If AS is a right S-poset then ((AS)
∗)∗ is called the second dual of AS
and denoted by (AS)
∗∗. If f is an S-morphism of right S-posets then (f∗)∗
will be denoted by f∗∗. If AS is a right S-poset and its second dual exists
then the mapping
ϕA : AS → hom(hom(AS , SS), SS) = (AS)
∗∗
defined by ϕA(a)(f) = f(a) for every a ∈ AS and f ∈ hom(AS , SS) is an
S-morphism of right S-posets. These S-morphisms determine a natural
transformation ϕ = (ϕA)AS∈POS-S ,
ϕ : IdPOS-S → hom(hom(−S , SS), SS).
In what follows ϕA will be called the natural S-morphism from AS into
(AS)
∗∗. Now the preliminaries are prepared to present our main definition.
Definition 2. An S-poset AS is called
(i) torsionless if ϕA is a monomorphism, i.e., kerϕA = ∆A,
(ii) regular torsionless if ϕA is a regular monomorphism, i.e.,
−→
kerϕA = ξA,
(iii) dense if ϕA is surjective,
(iv) reflexive if ϕA is an isomorphism, and
(v) regular reflexive if ϕA is an order isomorphism.
Notice that since ϕA(a) ∈ (AS)
∗∗ for every a ∈ AS , second duals exist
whenever duals exist. We start with two statements which are immediate
from the definition.
“adm-n3” — 2018/10/20 — 9:02 — page 81 — #87
R. Khosravi 81
Lemma 1. An S-poset AS, |AS | > 1, is regular torsionless (torsionless)
if and only if for every x, y ∈ AS , x � y(x 6= y), there exists f ∈
hom(AS , SS) such that f(x) � f(y) (f(x) 6= f(y)).
Lemma 2. The dual of the one-element right S-poset ΘS exists and ΘS
is regular torsionless if and only if S contains a left zero. In this case all
right S-posets have duals.
Lemma 3. The following hold for a pomonoid SS.
(i) SS is regular torsionless.
(ii) Every S-subposet of a regular torsionless S-poset is regular torsion-
less.
(iii) A retract of a regular torsionless S-poset is regular torsionless.
(iv) Let {Ai}i∈I be a family of regular torsionless S-posets. If
∐
i∈I Ai
is regular torsionless, then each Ai is regular torsionless.
(v) Let {Ai}i∈I be a family of regular torsionless S-posets , then
∏
i∈I Ai
is regular torsionless.
Proof. (i) If x, y ∈ SS , x � y, then for idS ∈ hom(SS,SS) we have
idS(x) � idS(y). So SS is regular torsionless by Lemma 1.
(ii) Let BS be an S-subposet of AS . If x, y ∈ BS , x
y, then there
exists f ∈ hom(AS , SS) such that f(x) � f(y). Now f |BS
∈ hom(BS , SS)
and f |BS
(x) � f |BS
(y). Hence BS is regular torsionless by Lemma 1.
(iii) and (iv) immediately follow from (ii).
(v) Let πi :
∏
i∈I Ai → Ai, i ∈ I, be the projections. If x = (xi)I , y =
(yi)I ∈
∏
i∈I Ai, x � y, then there exists j ∈ I such that xj � yj . Since Aj
is regular torsionless, there exists f ∈ hom(Aj , SS) such that f(xj) � f(yj).
Now fπj ∈ hom(
∏
i∈I Ai, SS) and (fπj)(x) � (fπj)(y), and the result
follows.
The following proposition shows that AS is regular torsionless if and
only if it can be embedded into some direct power of S.
Proposition 2. An S-poset AS is regular torsionless if and only if AS
can be embedded into SI for some non-empty set I.
Proof. Necessity. If AS is regular torsionless then ϕA : AS → (AS)
∗∗ is a
regular monomorphism. As we know (AS)
∗∗ = hom(hom(AS , SS), SS) can
be considered as an S-subposet of SI where I has the same cardinality as
the underlying set of the left S-poset hom(AS , SS), and so AS can be an
embedded into SI .
“adm-n3” — 2018/10/20 — 9:02 — page 82 — #88
82 On regular torsionless S-posets
Sufficiency. Suppose AS is an S-subposet of SI for some non-empty
set I. By Lemma 2, parts (i, ii, v), SS is regular torsionless, SI is regular
torsionless, and so AS is regular torsionless.
The following proposition shows that from an S-poset which has a
dual one can always construct a regular torsionless S-poset.
Proposition 3. If an S-poset AS has a dual, then AS/kerϕA is regular
torsionless. Moreover, kerϕA is the smallest congruence with this property,
i.e., if AS/ρ is regular torsionless for some congruence ρ on AS then
kerϕA 6 ρ.
Proof. Notice that for each x, y ∈ AS
(x, y) ∈
−→
kerϕA ⇐⇒ (∀f ∈ (AS)
∗)(f(x) 6 f(y)).
That is,
−→
kerϕA = ∩f∈A∗
−→
kerf . Suppose [a] � [b] in AS/kerϕA for a, b ∈ AS .
Then there exists f ∈ (AS)
∗ such that f(a) � f(b). Define a map-
ping f : AS/kerϕA → SS by f([x]) = f(x) for every x ∈ AS . Since
−→
kerϕA 6
−→
kerf , f is a well-defined S-morphism. As f([a]) = f(a) � f(b) =
f([b]), AS/kerϕA is regular torsionless by Lemma 1.
Moreover, suppose that AS/ρ is regular torsionless for a congruence ρ
on AS and x 6kerϕA
y, x, y ∈ AS . Let π : AS → AS/ρ be the canonical
epimorphism. Then (gπ)(x) 6 (gπ)(y) and thus g([x]ρ) 6 g([y]ρ) for
every homomorphism g : AS/ρ → SS . Since AS/ρ is regular torsionless,
Lemma 1 implies [x]ρ 6 [y]ρ, i.e. x 6ρ y. Hence kerϕA 6 ρ.
We use the notion A(I) for the amalgamated coproduct of two copies
of a pomonoid S over a proper right ideal I. The definition of A(I) for
S-posets appeared in [2] and in [7] it is proved that A(I) is a right S-poset.
Let I be a right ideal of a pomonoid S, x, y, z not belonging to S, and
A(I) = ({x, y} × (S \ I)) ∪ ({z} × I). Define a right S-action on A(I) by
(w, u)s =
{
(w, us) if us /∈ I, w ∈ {x, y}
(z, us) if us ∈ I.
The order on A(I) is defined as:
(w1, s) 6 (w2, t) ⇐⇒ (w1 = w2 and s 6 t)
or (w1 6= w2, s 6 i 6 t for some i ∈ I).
“adm-n3” — 2018/10/20 — 9:02 — page 83 — #89
R. Khosravi 83
In what follows we allow the core I also to be empty. This gives us
the coproduct of two copies S, as follows
S
∐
S = {(w, s)| s ∈ S,w ∈ {x, y}},
where (w1, s) 6 (w2, t) if and only if w1 = w2, s 6 t.
Proposition 4. Let AS = SS
∐
SS. Then
−→
kerϕA = ξA ∪ {((w1, u), (w2, v))|w1 6= w2 ∈ {x, y}, (∀s, t ∈ S)(su 6 tv)}.
Proof. Suppose that (a, b) ∈
−→
kerϕA. If a = (w, u) and b = (w, v), u, v ∈ S,
w ∈ {x, y}, a mapping f : SS
∐
SS → SS defined by f(c) = p if c =
(w, p), w ∈ {x, y} is a well-defined S-morphism and thus f(a) 6 f(b) by
the definition of
−→
kerϕA. Hence u 6 v, and (a, b) ∈ ξA. Now suppose that
a = (w1, u) and b = (w2, v), w1 6= w2, u, v ∈ S. Then for t, s ∈ S the
mappings f : SS
∐
SS → SS defined by
f(c) =
{
sp if c = (w1, p),
tp if c = (w2, p)
is a well-defined S-morphism. Since (a, b) ∈
−→
kerϕA, su = f(w1, u) 6
f(w2, v) = tv. Hence
−→
kerϕA ⊆ ξA ∪ {((w1, u), (w2, v))|w1 6= w2 ∈ {x, y}, (∀s, t ∈ S)(su 6 tv)}.
Now suppose that (w1, u), (w2, v) ∈ SS
∐
SS such that for each w1 6=
w2, s, t ∈ S, su 6 tv. Let f ∈ hom(AS , SS). It is easily checked that there
exist elements s, t ∈ S such that
f((w, p)) =
{
sp if w = w1,
tp if w = w2.
We have f((w1, u)) = su 6 tv = f((w2, v)) which means that
ξA ∪ {((w1, u), (w2, v))|w1 6= w2 ∈ {x, y}, (∀s, t ∈ S)(su 6 tv)} ⊆
−→
kerϕA.
Corollary 2. SS
∐
SS is regular torsionless if and only if
{((w1, u), (w2, v))|w1 6= w2 ∈ {x, y}, (∀s, t ∈ S)(su 6 tv)} = ∅.
“adm-n3” — 2018/10/20 — 9:02 — page 84 — #90
84 On regular torsionless S-posets
The following theorem is concerned with coproducts of regular torsion-
less right S-posets.
Theorem 1. The following assertions are equivalent for a pomonoid S:
(i) if Ai, i ∈ I, are regular torsionless right S-posets then
∐
i∈I Ai is
regular torsionless;
(ii) SS
∐
SS is regular torsionless;
(iii) for each u, v ∈ S there exist s, t ∈ S such that su � tv.
Proof. (i) → (ii) is obvious.
(ii) → (iii) follows from the previous corollary.
(iii) → (i). Suppose Ai, i ∈ I, are regular torsionless right S-posets
and let x, y ∈
∐
i∈I Ai such that x � y. If x, y ∈ Ai for some i ∈ I, then
there exists an S-morphism fi : Ai → SS such that fi(x) � fi(y). Define
f :
∐
i∈I Ai → SS by f |Aj
, j ∈ I, j 6= i, is an arbitrary S-morphism and
f |Ai
= fi. Clearly f is an S-morphism for which f(x) � f(y).
Otherwise, suppose that x ∈ Ai, y ∈ Aj , i, j ∈ I, i 6= j. Take an
S-morphism f :
∐
i∈I Ai → SS for which f |Ak
= fk, k ∈ I, is an
arbitrary S-morphism. If now fi(x) 6 fj(y) then, by assumption for
fi(x), fj(y) ∈ S there exist s, t ∈ S such that sfi(x) � tfj(y). Then define
g :
∐
i∈I Ai → SS for which g|Ak
= fk, k ∈ I, k 6= i, j, g|Ai
= sfi, and
g|Aj
= tfj . Obviously, g is an S-morphism for which g(x) � g(y). Thus
∐
i∈I Ai is regular torsionless by Lemma 1.
Proposition 5. Let S be a pomonoid, e ∈ S an idempotent such that e ∦ 1
and I = es. Then the amalgamated coproduct A(I) is regular torsionless.
Proof. Let a, b ∈ A(I) and a � b. If a = (w1, u), b = (w2, v), u, v ∈ S,
wi ∈ {x, y, z}, and u � v, then a mapping f : A(I) → SS defined by
f(w, s) = s, for w ∈ {x, y, z}, is a well-defined S-morphism and f(a) �
f(b). It suffices to check only the case that a = (w1, u) and b = (w2, v),
u 6 v, w1 6= w2 ∈ {x, y}. If e 6 1, then a mapping f : A(I) → SS defined
by
f(c) =
{
s if c = (w, s), w 6= w2,
es if c = (w2, s).
is a well-defined S-morphism. Now if u = f(a) 6 ev = f(b), then u 6
ev 6 v which means that a 6 b, a contradiction. If 1 6 e, f can be defined
analogously. Therefore f(a) � f(b) and A(I) is regular torsionless by
Lemma 1.
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R. Khosravi 85
Similar to the argument of the previous proposition, the following
result can be proved.
Proposition 6. Let S be a pomonoid, e ∈ S an idempotent and I =
(es] 6= S; the principal right poideal generated by e or I = [es) 6= S . Then
the amalgamated coproduct A(I) is regular torsionless.
Now, we turn our attention to cyclic S-posets.
Proposition 7. Let ρ 6= ∆S be a right congruence on a pomonoid S. The
cyclic right S-poset S/ρ is regular torsionless if and only if for all s, t ∈ S,
If s �ρ t, there exists u ∈ S such that us � ut and x 6ρ y, x, y ∈ S,
implies ux 6 uy.
Proof. Necessity. Suppose S/ρ is regular torsionless and s, t ∈ S so that
s �ρ t. Then there exists f ∈ hom(S/ρ, SS) such that f([s]ρ) � f([t]ρ), by
Lemma 1. Let u = f([1]). Then us = f([1])s = f([s]) � f([t]) = f([1])t =
ut. But for any x, y ∈ S, x 6ρ y, ux = f([1])x = f([x]) 6 f([y]) =
f([1])y = uy.
Sufficiency. By the assumption, if s �ρ t, there exists u ∈ S such
that us � ut and x, y ∈ S, x 6ρ y, implies ux 6 uy. Now a mapping
f : S/ρ → SS defined by f([x]) = ux is a well-defined S-morphism such
that f(s) � f(t). Thus S/ρ is regular torsionless.
Corollary 3. Let KS be a non-trivial convex right ideal of a pomonoid S.
The Rees factor S-poset S/KS is regular torsionless if and only if for every
s, t ∈ S, [s] � [t] implies the existence of u, z ∈ S, z a left zero, such that
us � ut and uk = z for every k ∈ KS.
Proof. Repeat the argument of the proof of Proposition 7 and take z =
f([k]), then uk = f([1])k = f([k]) = z for every k ∈ KS .
Lemma 4. Let S be a pomonoid. If all Rees factor S-posets of the form
S/[sS], s ∈ S, are regular torsionless, then for every right cancellable
element c of S there exist u, v ∈ S such that cu 6 1 6 cv.
Proof. Suppose c ∈ S is right cancellable if S = [cS] the result follows.
Otherwise, [cS] is a proper convex right ideal of S. Since the Rees factor
S-poset S/[cS] is regular torsionless and 1 /∈ [cS] without loss of generality
we can suppose that [1] � [ct] for some t ∈ S. So by Corollary 3, there exist
u, z ∈ S, z a left zero, such that u � uct and uk = z for every k ∈ [cS].
In particular, uct = uc = z. Since z is a left zero, one has uc = zc. Using
right cancellability of c we get u = z = uct, a contradiction.
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86 On regular torsionless S-posets
2. On homological classification
In this section we present some results on homological classification.
We start with questions when some property implies regular torsionless.
Then we give some necessary conditions for regular torsionless S-posets
to satisfy some flatness properties.
Theorem 2. The following are equivalent for a pomonoid S:
(i) every free right S-poset is regular torsionless.
(ii) every projective right S-poset is regular torsionless.
(iii) for each u, v ∈ S there exist s, t ∈ S such that su � tv.
Proof. (i) ⇐⇒ (ii) follows from Lemma 3 since every projective S-poset
is an S-subposet of free S-poset.
(i) ⇐⇒ (iii) follows from Theorem 1 since SS is regular torsionless
and every free right S-poset is a coproduct of copies of SS .
Lemma 5. If all principally weakly (po-)flat right S-posets over
a pomonoid S are regular torsionless then S contains at least two different
left zeros.
Proof. Since the one-element right S-poset is principally weakly flat and
coproducts of principally weakly flat S-posets are principally weakly flat, a
right S-poset AS = {a}
∐
{b} is regular torsionless where a, b are two zeros.
Then there exists f ∈ hom(AS , SS) such that f(a) � f(b) by Lemma 1.
But, being images of zeros, f(a) and f(b) are left zeros of S.
Now, we discuss the converse direction of two previous results.
Proposition 8. All regular torsionless right S-posets are po-torsionfree.
Proof. Suppose AS is regular torsionless. Then AS is isomorphic to an
S-subposet of SI for some non-empty set I by Lemma 3. But SS is po-
torsionfree, products of po-torsionfree S-posets are po-torsionfree and
S-subposets of po-torsionfree S-posets are po-torsionfree and so AS is
po-torsionfree.
Let S be a pomonoid. We recall from [5] that a finitely generated left S-
poset SB is called finitely definable (FD) if the S-morphism SΓ⊗B → BΓ
is order-embedding for all non-empty sets Γ.
Proposition 9. If all regular torsionless right S-posets are principally
weakly po-flat then all principal left ideals of S are finitely definable (FD).
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R. Khosravi 87
Proof. By Lemma 3, SI is regular torsionless for any non-empty set I,
and so principally weakly po-flat by the assumption. So SI ⊗ Ss → (Ss)Γ
is order-embedding for each s ∈ S. Using Proposition 2.4 of [5] it follows
that all principal left ideals of S are finitely definable.
Note that for every s, t ∈ S we have H(s, t) = {(as, a′t)| as 6 a′t}
and Ŝ(p, q) = {(u, v) ∈ D(S)| ∃w ∈ S, u 6 wp,wq 6 v}.
Proposition 10. If all regular torsionless right S-posets are weakly po-flat
then all principal left ideals of S are finitely definable and for every for
every s, t ∈ S, if Ss ∩ (St] 6= ∅, then H(s, t) is a subset of Ŝ(p, q) for
some (p, q) ∈ H(s, t).
Proof. The S-poset SI is regular torsionless for each non-empty set I and
so weakly po-flat by the assumption. Thus Theorem 2.7 of [5] implies the
result.
Recall from [5] that the set L(a, b) := {(u, v) ∈ D(S)| ua 6 vb} is a
left S-subposet of D(S), and the set l(a, b) := {u ∈ S| ua 6 ub} is a left
ideal of S. Moreover, a pomonoid S is called a left PSF pomonoid if all
principal left ideals of a pomonoid S are strongly flat.
Proposition 11. If all regular torsionless right S-posets are strongly flat
then
(i) for each idempotent e ∈ S there exists s ∈ S such that es ∦ 1 and e
is incomparable with 1,
(ii) for all (s, t) ∈ D(S), L(s, t) = ∅ or is a cyclic left S-poset,
(iii) for all (s, t) ∈ D(S), l(s, t) = ∅ or is a principal left ideal of S,
(iv) S is a right PSF pomonoid.
Proof. Suppose e ∈ S is an idempotent such that es is incomparable with 1
for each s ∈ S . Then for I = (eS] the amalgamated coproduct A(I) is
regular torsionless by Proposition 6. Similarly if e ∦ 1 Proposition 5 implies
that A(eS) is regular torsionless, and so strongly flat by the assumption
which is a contradiction by Lemma 2.4 of [7]. Thus S satisfies condition (i).
The S-posets SI for any non-empty set I are regular torsionless and so
strongly flat. Form Corollary 3.5 of [5] it follows that S satisfies conditions
(ii) and (iii). We have all principal right ideals of S are regular torsionless
by Lemma 3, and so strongly flat by the assumption. Thus S satisfies
conditions (iv).
Using Theorem 3.3 of [5], similar to the proof of the previous proposition
we imply two following results.
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88 On regular torsionless S-posets
Proposition 12. If all regular torsionless right S-posets satisfy Condi-
tion (P), then
(i) for each idempotent e ∈ S there exists s ∈ S such that es ∦ 1 and e
is incomparable with 1,
(ii) for all (s, t) ∈ D(S), L(s, t) = ∅ or is a cyclic left S-poset.
Proposition 13. If all regular torsionless right S-posets satisfy Condi-
tion (E), then for all (s, t) ∈ D(S), L(s, t) = ∅ or is a cyclic left S-poset.
A pomonoid S is called a rpp pomonoid if the S-subposet sS is pro-
jective for each s ∈ S.
Proposition 14. For a pomonoid S if all regular torsionless right S-posets
are projective, then
(i) S is po-cancellative,
(ii) incomparable principal right ideals of S are disjoint,
(iii) S satisfies the ascending chain condition (ACC) for principal right
ideals.
Proof. By Lemma 3, the S-poset SI is regular torsionless for each non-
empty set I and so projective by the assumption. Corollary 1 implies that
S is right po-cancellative. Using Lemma 3, all right ideals of S are regular
torsionless, and so projective by the assumption. So S is a rpp pomonoid
and satisfies conditions (ii) and (iii) from Theorem 2.8 of [8]. Suppose
sx 6 sy, s, x, y ∈ S. Since S is a rpp pomonoid there exists an idempotent
e ∈ S such that ex 6 ey. But since S is right po-cancellative, e = 1. Thus
x 6 y, and S is po-cancellative.
Acknowledgements
The author wish to express their appreciation to the anonymous
referees for their constructive comments improving the paper. She would
also like to acknowledge Professor V. Lyubashenko for providing the
communications.
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Contact information
R. Khosravi Department of Mathematics, College of Science,
Fasa University, Fasa, Iran.
P.O. Box 74617-81189
E-Mail(s): khosravi@fasau.ac.ir
Received by the editors: 08.08.2016
and in final form 30.06.2017.
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