Algebra and Discrete Mathematics RESEARCH ARTICLE
Volume 18 (2014). Number 2, pp. 234–249
© Journal “Algebra and Discrete Mathematics”
Morita equivalence for partially ordered
monoids and po-Γ-semigroups with unities
Sugato Gupta1 and Sujit Kumar Sardar
Communicated by V. A. Artamonov
Abstract. We prove that operator pomonoids of a po-Γ-
semigroup with unities are Morita equivalent pomonoids. Conversely,
we show that if L and R are Morita equivalent pomonoids then a
po-Γ-semigroup A with unities can be constructed such that left
and right operator pomonoids of A are Pos-isomorphic to L and R
respectively. Using this nice connection between po-Γ-semigroups
and Morita equivalence for pomonoids we, in one hand, obtain some
Morita invariants of pomonoids using the results of po-Γ-semigroups
and on the other hand, some recent results of Morita theory of
pomonoids are used to obtain some results of po-Γ-semigroups.
Introduction and notations
A monoid S together with a partially ordered relation 6 on it is called a
partially ordered monoid (in short pomonoid) if s 6 t implies us 6 ut and
su 6 tu for all s, t, u ∈ S. Let S be a pomonoid with identity 1. A poset
(A,6) together with a mapping A×S → A, denoted (a, s) 7→ as, is called
a right S-poset if for all a, b ∈ A and s, t ∈ S we have (1) a(st) = (as)t,
(2) a1 = a, (3) a 6 b implies as 6 bs, and (4) s 6 t implies as 6 at. Left
S-posets can be defined analogously. We shall distinguish left and right
S-posets by writing SA and AS respectively. If A is simultaneously a left
1The first author is grateful to University Grants Commission, Govt of India, for
providing research support as JRF.
2010 MSC: 20M50, 06F05.
Key words and phrases: Morita equivalence for pomonoids, Morita invariant,
Morita context, Po-Γ-semigroup.
S. Gupta, S. K. Sardar 235
T -poset and a right S-poset such that (ta)s = t(as) for all a ∈ A, t ∈ T
and s ∈ S then we call A an T -S-biposet and denote it by TAS . Let A
and B be two right S-posets. Then a mapping f : A→ B is called a right
S-poset morphism if for all a, b ∈ A and s ∈ S we have (1) f(as) = f(a)s
and (2) a 6 b implies f(a) 6 f(b). Left S-morphisms can be defined
analogously. If A and B be two T -S-biposets then a mapping f : A→ B
is called a T -S-biposet morphism if f is simultaneously a left T -poset
morphism and a right S-poset morphism. The category formed by right
S-posets together with the right S-morphisms is denoted by PosS . Its left
analogue is denoted by SPos. Also the category of T -S-biposets is denoted
by TPosS . In the category PosS , SPos and TPosS the morphism sets
are denoted by PosS(A,B), SPos(A,B) and TPosS(A,B) respectively.
These categories are enriched over the category Pos of posets (with order
preserving maps as morphisms), i.e., the morphism sets are posets with
respect to pointwise order. Again a Pos-functor between such categories
is a functor that preserves the order of morphisms.
Two pomonoids S and T are said to be Morita equivalent if the
categories PosS and PosT are two Pos-equivalent categories. Let A be a
right S-poset and B be a left S-poset. Denote by A× B the Cartesian
product of sets A and B. Then A × B is a partially ordered set in the
Cartesian order (a, b) 6 (c, d) if and only if a 6 c and b 6 d. Put
H = {((as, b), (a, sb)) | a ∈ A, b ∈ B, s ∈ S} and let ρ = ρ(H) be
the congruence generated by H over A × B. The quotient (A × B)/ρ
is a partially ordered set, called the tensor product of A and B over
S, denoted by A ⊗S B. As usual we denote the equivalence class of
(a, b) in A ⊗S B by a ⊗ b. The order relation on A ⊗S B is defined as
follows: a⊗ b 6 a′ ⊗ b′ in A⊗S B if and only if there exist a1, a2, ..., an ∈
A, b2, ..., bn ∈ B, s1, t1, ..., sn, tn ∈ S such that
a 6 a1s1,
a1t1 6 a2s2, s1b 6 t1b2,
a2t2 6 a3s3, s2b2 6 t2b3,
... ...
antn 6 a′, snbn 6 tnb′.
A six-tuple 〈S, T,S PT ,T QS , τ, µ〉 is said to be a Morita context where
S and T are pomonoids, SPT ∈ SPosT , TQS ∈ TPosS , and τ : S(P ⊗T
Q)S → SSS and µ : T (Q⊗S P )T → TTT are biposet morphisms such
that for all p, p′ ∈ P and q, q′ ∈ Q we have τ(p ⊗ q)p′ = pµ(q ⊗ p′) and
qτ(p⊗ q′) = µ(q⊗ p)q′. For more notions of Morita equivalent pomonoids
reader is referred to [11,12,20–23].
236 Morita Equivalence for Pomonoids
If A and Γ be two non-empty sets then A is said to be a Γ-semigroup
if there exist mappings A × Γ × A → A, denoted by (a, γ, b) 7→ aγb,
and Γ × A × Γ → Γ, denoted by (α, a, β) 7→ αaβ, satisfying (aαb)βc =
a(αbβ)c = aα(bβc) for all a, b, c ∈ A and α, β ∈ Γ. For a natural example
of a Γ-semigroup let A := Maps(M,N) and Γ := Maps(N,M) where
M and N are two non-empty sets. Then A is a Γ-semigroup where aγb
and αaβ are usual mapping compositions. If A is a Γ-semigroup and ρ is
the equivalence relation on Γ×A defined by (α, a)ρ(β, b) if and only if
xαa = xβb and αaγ = βbγ for all x ∈ A and γ ∈ Γ, then we denote the
equivalence class of (α, a) by [α, a]. Now the right operator semigroup of the
Γ-semigroup A is defined to be R = (Γ×A)/ρ = {[α, a] | a ∈ A,α ∈ Γ},
where the composition is defined by [α, a][β, b] = [α, aβb]. The left operator
semigroup L is defined analogously. We note here that A is a L−R-biset.
Now if there exists an element [γ, f ] in R such that xγf = x for all x ∈ A,
then it is called the right unity of A. We note that [γ, f ] becomes the
identity of R. Similarly the left unity of A is defined. A Γ-semigroup
is said to be a Γ-semigroup with unities if it has both left and right
unities. A Γ-semigroup A is said to be a partially ordered Γ-semigroup (in
short po-Γ-semigroup) if (1) A and Γ are posets, (2) a 6 b in A implies
that aαc 6 bαc, cαa 6 cαb in A and γaα 6 γbα in Γ for all a, b, c ∈ A
and α, γ ∈ Γ, (3) α 6 β in Γ implies that αaγ 6 βaγ, γaα 6 γaβ
in Γ and aαb 6 aβb in A for all a, b ∈ A and α, β, γ ∈ Γ. It is well
known that operator semigroups of a (po)Γ-semigroup with unities are
(po)monoids. For more details of Γ-semigroups, operator semigroups and
po-Γ-semigroups we refer respectively to [14,15], [4] and [8].
The notion of Γ-semigroup was introduced by M.K. Sen [18, 19] in
1981 as a generalization of semigroup. Subsequently Dutta and Adhikari
generalized partially ordered semigroups to po-Γ-semigroups with unities
in [8] in the year 2004. They also introduced the notion of operator
pomonoids associated with a po-Γ-semigroup. Among others, Knauer [10]
and Banaschewski [2], contributed a lot to develop the theory of Morita
equivalence of monoids. Recently Laan [12] developed a theory of Morita
equivalent partially ordered monoids or shortly pomonoids and obtained
Morita I, Morita II and Morita III. The Morita theory of posemigroups
and pomonoids are further investigated by Tart [22,23] and Targla and
Laan [21].
Using mainly the works of Knauer [10] and Banaschewski [2], in [16] we
could find a close connection between the Morita equivalence of monoids
and Γ-semigroups with unities which was used to enrich each other’s
study. As a sequel to this and motivated by the works of Laan [12],
S. Gupta, S. K. Sardar 237
Tart [22,23], Targla et al. [21] on Morita theory of pomonoids we make
an attempt here to find connection between po-Γ-semigroups and Morita
equivalent pomonoids. In this venture we show that operator pomonoids
of a po-Γ-semigroup with unities are Morita equivalent and conversely if
L and R are two Morita equivalent pomonoids then we can construct a
po-Γ-semigroup with unities whose left and right operator pomonoids are
Pos-isomorphic [12] to L and R respectively. As an application of these
results we have been able to apply the existing results of po-Γ-semigroups
to extend the Morita theory of pomonoids in the form of obtaining some
properties of pomonoids which remain invariant under Morita equivalence.
In the opposite direction we have used some recent results of Morita
theory of pomonoids due to Tart [22, 23] and Targla and Laan [21] to
obtain some results of po-Γ-semigroups.
1. Relating po-Γ-semigroup with Morita equivalence
In this section we are going to explore the relationship between po-
Γ-semigroup with unities and Morita equivalence for pomonoids. Let us
first obtain the following lemma.
Lemma 1. Let A be a po-Γ-semigroup with unities and its left and right
operator pomonoids be respectively L and R. Then LAR and RΓL are
biposets.
Proof. By definition of po-Γ-semigroup, A and Γ both are posets. We
define
L×A→ A and A×R→ A by
[a, α]c := aαc and c[β, b] := cβb respectively.
Again, L and R are pomonoids with respect to the ordering defined below:
(i) On L, [a, α] 6 [b, β] if and only if aαc 6 bβc and γaα 6 γbβ for all
c ∈ A and γ ∈ Γ ;
(ii) On R, [α, a] 6 [β, b] if and only if cαa 6 cβb and αaγ 6 βbγ for all
c ∈ A and γ ∈ Γ.
238 Morita Equivalence for Pomonoids
Now using the fact that A is a po-Γ-semigroup and L and R are
partially ordered monoids we deduce the following:
(1) For all a ∈ A and [α, b], [β, c] ∈ R, a([α, b][β, c]) = a[α, bβc] =
aα(bβc) = (aαb)βc = (a[α, b])[β, c].
(2) If [γ, f ] is the right unity of A then for all a ∈ A we have a[γ, f ] = a.
(3) For all [α, c] ∈ R, a 6 b gives aαc 6 bαc whence a[α, c] 6 b[α, c].
(4) For all a ∈ A, [α, b] 6 [β, c] gives aαb 6 aβc whence a[α, b] 6 a[β, c].
Hence A is a right R-poset. Similarly, A is a left L-poset. That A is a
biposet follows from the associative property of the po-Γ-semigroup A.
Similarly we can prove that RΓL is a biposet.
In the following theorem we obtain the Morita equivalence of the left
and right operator pomonoids of a po-Γ-semigroup with unities.
Theorem 1. Let A be a po-Γ-semigroup with unities and its left and right
operator pomonoids be respectively L and R. Then the following hold:
(1) L and R are Morita equivalent,
(2) A⊗R Γ ∼= L and Γ⊗L A ∼= R,
(3) LA,AR, RΓ and ΓL are cyclic projective generators of their respective
categories,
(4) L ∼= End(AR) ∼= End(RΓ) and R ∼= End(ΓL) ∼= End(LA).
Proof. By Lemma 1, L and R are pomonoids, LAR and RΓL are biposets.
So we define
τ : A⊗ Γ → L and µ : Γ⊗A→ R
as follows
τ(a⊗ α) = [a, α] and µ(α⊗ a) = [α, a].
Let a⊗ α = b⊗ β, for a, b ∈ A and α, β ∈ B. Then by the Theorem 5.2
of [20], there exist a1, a2, ..., an, b1, b2, ..., bm ∈ A, α2, ..., αn, β2, ..., βm ∈ B,
s1, t1, ..., sn, tn, u1, v1, ..., um, vm ∈ R such that,
a 6 a1s1, b 6 b1u1,
a1t1 6 a2s2, s1α 6 t1α2, b1v1 6 b2u2, u1β 6 v1β2,
a2t2 6 a3s3, s2α2 6 t2α3, b2v2 6 b3u3, u2β2 6 v2β3,
... ... ... ...
antn 6 b, snαn 6 tnβ; bmvm 6 a, umβm 6 vmα.
S. Gupta, S. K. Sardar 239
Now we again use the Lemma 1 and obtain
aαc 6 (a1s1)αc = a1(s1α)c
6 a1(t1α2)c = (a1t1)α2c
6 (a2s2)αc = a2(s2α)c
. . . . . .
6 an(tnβ)c = (antn)βc
6 bβc.
(1)
Similarly we have bβc 6 aαc. Since by definition A has partial ordering,
so aαc 6 bβc and bβc 6 aαc together implies aαc = bβc for any c ∈ A.
Similarly we can show that γaα = γbβ for any γ ∈ Γ. Hence [a, α] = [b, β].
Thus we see that the mapping τ is well-defined. Also it is clear that τ is
onto.
Again, let a ∈ A, α ∈ Γ and [b, β] ∈ L. Then we see that
τ([b, β](a⊗ α)) = τ([b, β]a⊗ α)
= τ(bβa⊗ α)
= [bβa, α]
= [b, β][a, α]
= [b, β]τ(a⊗ α).
Similarly τ((a⊗ α)[b, β]) = τ(a⊗ α)[b, β]. So τ preserves the action from
both sides.
Let a ⊗ α 6 b ⊗ β, for a, b ∈ A and α, β ∈ B. Then using Lemma 1
along with the definition of 6, we apply a similar argument, as given in
equation (1), and prove that aαc 6 bβc for all c ∈ A. Similarly we deduce
that γaα 6 γbβ for all γ ∈ Γ. Hence we obtain [a, α] 6 [b, β].
Then τ preserves the ordering. Hence τ becomes a surjective biposet
morphism. By a similar argument we see that µ is a surjective biposet
morphism.
Now we see that for all a, b ∈ A, α, β ∈ Γ,
τ(a⊗ α)b = [a, α]b = aαb = a[α, b] = aµ(α⊗ b) and
ατ(a⊗ β) = α[a, β] = αaβ = [α, a]β = µ(α⊗ a)β.
Hence 〈L,R,LAR,R ΓL, τ, µ〉 is a Morita context with τ and µ surjective.
Hence in view of Theorem 6 and Proposition 7 of [12] we obtain the
desired results namely (1)–(4).
240 Morita Equivalence for Pomonoids
Remark 1. Since we have deduced that operator pomonoids of a po-
Γ-semigroup A with unities are Morita equivalent so the examples of
non-isomorphic operator pomonoids will also become examples of non-
isomorphic Morita equivalent pomonoids.
To prove the converse part of the above theorem we first obtain some
characterization of two Morita equivalent pomonoids. In this regard we
prove the following lemma.
Lemma 2. Let MS be a right S-poset. Then MS ∼= PosS(S,MS).
Proof. For each m ∈MS , if we define
ρm : SSS →MS by ρm(s) := ms
then ρm ∈ PosS(S,MS). Now we define
ρ : MS → PosS(S,MS) by ρ(m) := ρm.
Then it will take more than a glance to see that ρ is an isomorphism.
Now we obtain the following characterization for Morita equivalent
pomonoids.
Theorem 2. Let S and T be two Morita equivalent pomonoids via inverse
Pos-equivalences F : PosS → PosT and G : PosT → PosS. Set P =
F (S) and Q = G(T ). Then P and Q are biposets SPT and TQS such that
(1) PT , QS are respectively cyclic projective generators for PosT and
PosS;
(2) T ∼= End(QS) and S ∼= End(PT );
(3) F ∼= PosS(QS , ) and G ∼= PosT (PT , );
(4) SPT ∼= PosS(Q,S) and TQS ∼= PosT (P, T ).
Proof. By Theorem 4 of [12], P and Q are biposets SPT and TQS . Also
in view of Theorem 3 of [12], (1) and (2) follow.
Now by Lemma 2, for any right S-poset MS , MS ∼= PosS(S,MS).
Then
G(MT ) ∼= PosS(S,G(MT )) ∼= PosT (F (S),MT ) ∼= PosT (PT ,MT ).
So we obtain G ∼= PosT (PT , ). Using this on T we deduce that
Q = G(T ) ∼= PosT (PT , T ).
Similarly we can prove F ∼= PosS(QS , ) and SPT ∼= PosS(Q,S). Hence
the theorem.
S. Gupta, S. K. Sardar 241
Now we can obtain the following theorem which gives the converse of
Theorem 1(1).
Theorem 3. Let L and R be two Morita equivalent pomonoids. Then
there exists a po-Γ-semigroup with unities whose left and right operator
pomonoids are Pos-isomorphic to L and R respectively.
Proof. Since L and R are Morita equivalent pomonoids, the categories
PosL and PosR are Pos-equivalent via inverse Pos-equivalences, say,
F : PosL → PosR and G : PosR → PosL.
Let A = F (L) and Γ = G(R). Then by Theorem 2 we see that
(1) LAR and RΓL are biposets;
(2) AR and ΓL are respectively cyclic projective generators for PosR
and PosL;
(3) R ∼= End(ΓL) and L ∼= End(AR);
(4) F ∼= PosL(ΓL, ) and G ∼= PosR(AR, );
(5) LAR ∼= PosL(Γ, L) and RΓL ∼= PosR(A,R).
Now considering Γ as PosR(A,R) we define the mappings
A× Γ×A→ A and Γ×A× Γ → Γ
such that for a, b, x ∈ A and α, β, γ ∈ Γ
(a, γ, b) 7→ a(γ(b)) and (α, x, β) 7→ (α(x))β.
As a consequence, we deduce the following equalities:
(aαb)βc = (a(α(b)))(β(c)) = a(α(b)β(c)), since A is a right R-poset;
aα(bβc) = a(α(b(β(c)))) = a(α(b)β(c)), since α is a right R-poset
morphism;
a(αbβ)c = a(((α(b))β)(c)) = a(α(b)β(c)), since PosR(A,R) is a left
R-poset.
Consequently, A becomes a Γ-semigroup. Let L′ and R′ be the left
and right operator semigroups of the Γ-semigroup A. Then in order to
complete the proof we consider the case of R and R′ as the case of L and
L′ will follow in a similar fashion. We define
f : R′ → R by f([α, a]) = α(a).
242 Morita Equivalence for Pomonoids
Then
[α, a] = [β, b] implies that αaγ = βbγ for all γ ∈ Γ
i.e., (α(a))γ = (β(b))γ for all γ ∈ Γ
i.e., α(a) = β(b).
Hence the mapping f is well-defined. Again,
α(a) = β(b) implies that (α(a))γ = (β(b))γ and x(α(a)) = x(β(b))
for all x ∈ A, γ ∈ Γ
i.e., αaγ = βbγ and xαa = xβb
for all x ∈ A, γ ∈ Γ
i.e., [α, a] = [β, b].
Hence f is injective. Again, since AR is a generator of PosR, there exists a
right R-poset epimorphism ψ : A→ R (see [11]) such that for any r ∈ R,
there exists a ∈ A with ψ(a) = r. Then we have f([ψ, a]) = r. Hence the
mapping f is surjective.
Also for all a, b ∈ A and α, β ∈ Γ,
f([α, a][β, b]) = f([α, aβb]) = α(a(β(b)))
= α(a)β(b) = f([α, a])f([β, b]).
Hence f is a semigroup morphism. Consequently, R′ and R are isomorphic
as monoids.
Hence A is a Γ-semigroup with unities whose left and right operator
semigroups are isomorphic to L and R respectively. Now in view of
Lemma 1 and the fact that Γ ∼= PosR(A,R) has pointwise ordering, we
deduce that A is a po-Γ-semigroup.
Now in order to complete the proof we show that R′ is Pos-isomorphic
to R which we accomplish by proving that the isomorphism, defined
above, preserves the orders. Now, if the right unity of A is [δ, e] then for
[α, a], [β, b] ∈ R′, [α, a] 6 [β, b] implies that αaγ 6 βbγ for all γ ∈ Γ. So,
in particular,
αaδ 6 βbδ
i.e., (α(a))δ 6 (β(b))δ
i.e., ((α(a))δ)(e) 6 ((β(b))δ)(e) as RΓL ∼= PosR(A,R)
i.e., (α(a))(δ(e)) 6 (β(b))(δ(e)) as RΓL ∼= PosR(A,R)
i.e., α(a) 6 β(b) as δ(e) becomes the identity of R.
S. Gupta, S. K. Sardar 243
Again,
f([α, a]) 6 f([β, b]) implies that α(a) 6 β(b)
i.e., c(α(a)) 6 c(β(b)) for all c ∈ A
and (α(a))γ 6 (β(b))γ for all γ ∈ Γ
i.e., cαa 6 cβb for all c ∈ A
and αaγ 6 βbγ for all γ ∈ Γ
i.e., [α, a] 6 [β, b].
Consequently, f , f−1 preserve the ordering. This completes the proof.
We conclude this section by combining Theorems 1 and 3 into the
following theorem.
Theorem 4. Two pomonoids L and R are Morita equivalent if and only
if there exists a po-Γ-semigroup with unities whose operator pomonoids
are isomorphic to L and R.
2. Applications
In this section we find some applications of Theorem 4. In fact by this
theorem and some well-known results of po-Γ-semigroups we obtain some
properties of pomonoids which remain invariant under Morita equivalence.
Remark 2. In this section, what we use as the results of po-Γ-semigroup,
they are actually taken without proof from their counterparts in Γ-
semigroup [1, 4–7] because their proofs can be accomplished by slight
modification of the proofs of their analogues.
Theorem 5. Let L and R be Morita equivalent pomonoids. Then there
exists an inclusion preserving bijection between the set of all ideals of R
and the set of all ideals of L.
Proof. By Theorem 3, there exists a po-Γ-semigroup A with unities whose
left and right operator pomonoids L1 and R1 are isomorphic to L and R
respectively. So it is sufficient to prove the result for L1 and R1. Now for
each P ⊆ L1 and M ⊆ R1, we define (see [4])
P+ = {x ∈ A | [x, α] ∈ P for all α ∈ Γ} and
M∗ = {x ∈ A | [α, x] ∈M, for all α ∈ Γ}.
244 Morita Equivalence for Pomonoids
Also for each Q ⊆ A, we define (see [4])
Q+′ = {[x, α] ∈ L1 | xαa ∈ Q, for all a ∈ A} and
Q∗′ = {[α, x] ∈ R1 | aαx ∈ Q, for all a ∈ A}.
Then there exists an inclusion preserving bijection between the set of all
ideals of A and the set of all ideals of L1 (see [4] and Remark 2) via the
mapping
Q 7→ Q+′ with the inverse mapping P 7→ P+.
And also there exists an inclusion preserving bijection between the set of
all ideals of A and the set of all ideals of R1 via the mapping
Q 7→ Q∗′ with the inverse mapping M 7→M∗.
Then we obtain the desired inclusion preserving bijection between the
set of all ideals of R1 and the set of all ideals of L1 via the composition
mapping given by
J 7→ J∗ 7→ (J∗)+′ with the corresponding inverse I 7→ I+ 7→ (I+)∗′ .
Now from [1, 7, 8] and in view of Remark 2, we know that all the
mappings +, +′, ∗ and ∗′ carry prime ideals to prime ideals; weakly
prime ideals to weakly prime ideals; semiprime ideals to semiprime ideals;
primary ideals to primary ideals; semiprimary ideals to semiprimary
ideals; nil ideals to nil ideals and nilpotent ideals to nilpotent ideals. Then
by the same argument as applied in the above theorem we see that the
composition mappings
J 7→ J∗ 7→ (J∗)+′ and its inverse I 7→ I+ 7→ (I+)∗′ .
are respectively mappings from R1 to L1 and L1 to R1 carrying prime ide-
als to prime ideals; weakly prime ideals to weakly prime ideals; semiprime
ideals to semiprime ideals; nil ideals to nil ideals and nilpotent ideals to
nilpotent ideals. This gives rise to the following result.
Theorem 6. Let L and R be Morita equivalent pomonoids. Then there
exists an inclusion preserving bijection between the set of all prime (weakly
prime, semiprime, primary, semiprimary, nil, nilpotent) ideals of R and
the set of all prime (weakly prime, semiprime, primary, semiprimary, nil,
nilpotent) ideals of L.
S. Gupta, S. K. Sardar 245
Also using Remark 2 and combining the respective results of [5, 7] for
left operator L and right operator R of a po-Γ-semigroup A we find that
different types of radicals viz. prime radical, Schwarz radical, Clifford
radical are also preserved by the mappings
J 7→ J∗ 7→ (J∗)+′ and its inverse I 7→ I+ 7→ (I+)∗′ .
Hence we obtain the following result.
Theorem 7. Let L and R be Morita equivalent pomonoids. Then the
mapping J 7→ J∗ 7→ (J∗)+′ (I 7→ I+ 7→ (I+)∗′) carries prime (Schwarz,
Clifford) radical of R (L) to prime (Schwarz, Clifford) radical of L (re-
spectively R).
Theorem 8. Let L and R be Morita equivalent pomonoids. Then L is
Noetherian if and only if R is Noetherian.
Proof. By Theorem 3, there exists a po-Γ-semigroup A with left and right
unities whose left and right operator pomonoids L1 and R1 are isomorphic
to L and R respectively. According to [6] and Remark 2, L1 is Noetherian
if and only if A is Noetherian if and only if R1 is Noetherian. Hence the
result.
In a similar way by using the analogous results of [1] and in view of
Remark 2, we get the following theorem.
Theorem 9. Let L and R be Morita equivalent pomonoids. Then L is
primary (semiprimary) if and only if R is primary (semiprimary).
Theorem 10. Let L and R be Morita equivalent pomonoids. Then there
exists an inclusion preserving bijection between the set of all ordered
semilattice congruence (ordered fuzzy semilattice congruence) of R and
the set of all ordered semilattice congruence (respectively, ordered fuzzy
semilattice congruence) of L.
Proof. By Theorem 3, there exists a po-Γ-semigroup A with unities whose
left and right operator pomonoids L1 and R1 are isomorphic to L and R
respectively. So it is sufficient to prove the result for L1 and R1. Now for
each relation σ on L1 and ω on R1, we define (see [17]) relations σ+ and
ω∗ respectively on A as follows:
xσ+y if and only if [x, α]σ[y, α] for all α ∈ Γ and
xω+y if and only if [α, x]σ[α, y] for all α ∈ Γ.
246 Morita Equivalence for Pomonoids
Also for each relation ρ on A, we define (see [17]) relations ρ+′ on L1 and
ρ∗′ on R1 as follows:
[x, α]ρ+′ [y, β] if and only if (xαa)ρ(yβa) for all a ∈ A and
[α, x]ρ∗′ [β, y] if and only if (aαx)ρ(aβy) for all a ∈ A.
Then there exists an inclusion preserving bijection between the set of
all ordered semilattice congruence (ordered fuzzy semilattice congruence)
of A and the set of all ordered semilattice congruence (ordered fuzzy
semilattice congruence) of L1 (see [17]) via the mapping
ρ 7→ ρ+′ with the inverse σ 7→ σ+.
And also there exists an inclusion preserving bijection between the set of
all ordered semilattice congruence (ordered fuzzy semilattice congruence)
of A and the set of all ordered semilattice congruence (ordered fuzzy
semilattice congruence) of R1 via the mapping
ρ 7→ ρ∗′ with the inverse ω 7→ ω∗.
Then we find an inclusion preserving bijection between the set of all
ordered semilattice congruence (ordered fuzzy semilattice congruence)
of R1 and the set of all ordered semilattice congruence (ordered fuzzy
semilattice congruence) of L1 via the composition mapping
ω 7→ ω∗ 7→ (ω∗)+′ with the inverse σ 7→ σ+ 7→ (σ+)∗′ .
So far, in this section, we have used the results of po-Γ-semigroups
to obtain results of pomonoids viz. the Morita invariants. Now in the
rest of this section we move the other way round i.e., we use results of
posemigroups or pomonoids obtained by Tart [22, 23] and Targla and
Laan [21] to obtain some results of po-Γ-semigroups.
In what follows A denotes a po-Γ-semigroup with unities and L and
R respectively denote the left and right operator pomonoids of A. Then
in view of Theorem 1, L and R are Morita equivalent pomonoids. So the
following result follows easily from Proposition 6.1 of [22].
Theorem 11. (1) L is regular if and only if R is regular.
(2) Sets of regular D-classes of L and R have same cardinalities.
S. Gupta, S. K. Sardar 247
Theorem 12. If the order on L is either total, discrete, directed or
a semiorder, then the order on R is also total, discrete, directed or a
semiorder and vice-versa.
Proof. Since L and R are Morita equivalent pomonoids, they are strongly
Morita equivalent posemigroups with common local units. Hence the
result follows from Proposition 3.2 of [23].
Theorem 13. (1) If L satisfies an inequality, then R also satisfies the
same inequality and vice-versa. Moreover if L satisfies an identity,
then R also satisfies the same identity and vice-versa.
(2) If L is commutative or a band or a semilattice then R is also so
and vice-versa.
(3) Greatest commutative images of L and R are isomorphic.
Proof. Since L and R are strongly Morita equivalent posemigroups with
common two-sided weak local units, (1), (2) and (3) are just consequences
of Theorem 3.1, Corollary 3.2 and Theorem 3.2 of [23] respectively.
By a similar argument, the following theorem follows from our Theo-
rem 1 and Theorem 4.1 of [23].
Theorem 14. There is a lattice isomorphism between the lattices of
ideals (downwards closed ideals, upwards closed ideals, convex ideals) of
L and R which takes finitely generated ideals to finitely generated ideals
and principal ideals to principal ideals.
Theorem 15. (1) Congruence lattices of L and R are isomorphic.
(2) Picard groups of L and R are isomorphic.
Proof. Since L andR are Morita equivalent pomonoids, (1) and (2) are just
consequences of Corollary 1 of [21] and Corollary 5 of [12] respectively.
Concluding remark
(1) The results of the last section illustrates that Theorem 4, which
connects closely Morita equivalence of pomonoids and po-Γ-semigroups,
can supplement the study of each other.
(3) It will be nice if the results obtained here for pomonoids still hold
in partially ordered semigroups (without identity).
248 Morita Equivalence for Pomonoids
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Contact information
S. Gupta,
S. K. Sardar
Department of Mathematics, Jadavpur Univer-
sity, Kolkata, India
E-Mail(s): sguptaju@gmail.com,
sksardarjumath@gmail.com
Received by the editors: 30.01.2013
and in final form 03.05.2013.