Lattices and quantales of ideals of semigroups and their preservation under Morita contexts

. We study properties of the lattice of unitary ideals of a semigroup. In particular, we show that it is a quantale. We prove that if two semigroups are connected by an acceptable Morita context then there is an isomorphism between the quantales of unitary ideals of these semigroups. Moreover, factorisable ideals corresponding to each other under this isomorphism are strongly Morita equivalent.


Introduction
Just like in rings, Morita contexts have turned out to be very useful for studying Morita equivalence of semigroups as well. Using Morita contexts it is possible to prove that many properties are shared by all semigroups in the same Morita equivalence class (such properties are said to be Morita invariant). In our earlier paper [4] we proved that the ideal lattices of Morita equivalent semigroups with weak local units are isomorphic. However, the question if this holds also for larger classes of semigroups remained open. In the present article we show that the ideal lattices of Morita equivalent firm semigroups need not be isomorphic. On the other hand, it turns out that considering unitary ideals instead of all ideals it is possible to prove the isomorphism of the corresponding lattices (actually even quantales) under much weaker assumptions-putting no Then T is an ideal of the semigroup S and we have a Morita context with homomorphisms where φ is surjective, but θ is not. However, this context is acceptable. Definition 1.5. Semigroups S and T are said to be • strongly Morita equivalent [10] if they are contained in a unitary surjective Morita context, • Morita equivalent if the category of firm right S-acts is equivalent to the category of firm right T -acts.
Strongly Morita equivalent semigroups are necessarily factorisable. In fact, on the class of factorisable semigroups Morita equivalence coincides with strong Morita equivalence [5,Theorem 4.11].

The quantale of unitary ideals
In this section we study the lattice of unitary ideals of a semigroup. Some results will be formulated in the more general situation of unitary biacts.
We denote the set of all unitary sub-biacts of a biact S X T by USub( S X T ). This is a poset with respect to inclusion where the empty sub-biact is the bottom element. Every biact S X T has also a largest unitary sub-biact: the union of all its unitary sub-biacts, which will be denoted by U( S X T ) or just U(X). Similar considerations hold for right (or left) acts.

Lemma 2.1.
An element x belongs to U( S X T ) if and only if there exist elements s i ∈ S, t i ∈ T , and x i , y i ∈ X such that Proof. Let V denote the set of all elements x for which the above equalities hold. Then the inclusion U( S X T ) ⊆ V holds because U( S X T ) is unitary.
Conversely, consider an arbitrary element x ∈ V with the above equalities. For every s ∈ S and t ∈ T we have sx ∈ V because of the equalities and xt ∈ X because of the equalities The equality V = V T follows from the fact that x = (s 1 y 1 )t 1 and s 1 y 1 ∈ V because of the equalities Similarly, we have V = SV because x = s 1 x 1 and x 1 ∈ V due to Since the union of unitary sub-biacts is also a unitary sub-biact, the poset USub( S X T ) is a complete lattice where joins are unions. If A i , i ∈ I, are unitary subacts of S X T then it can be checked that Proof. For any unitary sub-biacts A, B, C ∈ USub( S X T ) we have to prove the .
In particular, every y i belongs to B or C. Note also that, for every n ∈ N, we have There are two possibilities.
(1) The set {i ∈ N | y i ∈ B} is infinite. Take an arbitrary k ∈ N. Then there exists l ∈ N such that y k+l ∈ B. Now implies that y k ∈ B, because B is a sub-biact of S X T . Also x k ∈ B because x k = y k t k , and thus x k , y k ∈ A ∩ B. By Lemma 2.1 this means that x ∈ U(A ∩ B). (2) The set {i ∈ N | y i ∈ C} is infinite. Then a similar argument shows that x ∈ U(A ∩ C).
A semigroup S is said to have weak local units if, for every s ∈ S, there exist u, v ∈ S such that s = us = sv. Proposition 2.5. If S and T are semigroups with weak local units and S X T is a biact then the lattice USub( S X T ) is algebraic.
Proof. As mentioned before, this lattice is complete. It is straightforward to check that the compact elements of the lattice USub( S X T ) are the finitely generated unitary sub-biacts of S X T . If A ∈ USub( S X T ) then each element a of A can be written as a = sa = at for some s ∈ S, t ∈ T , hence the 1-generated sub-biact S 1 aT 1 is unitary. Now A is the join of the compact elements S 1 aT 1 , a ∈ A. Now we turn our attention to unitary ideals. Definition 2.6. We say that an ideal I of a semigroup S is right (resp. left) unitary if the S-act I S (resp. S I) is unitary, that is, IS = I (resp. SI = I).

An ideal is called unitary if it is both right and left unitary.
We denote the set of all unitary ideals of a semigroup S by UId(S). Every semigroup has the unitary ideal ∅. If S is factorisable then S is also a unitary ideal of itself. If a semigroup has a zero element 0 then {0} is a unitary ideal. From the two-sided analogue of Proposition 2.4 of [9] we have the following. Actually, even more is true.

Proposition 2.9. A semigroup has weak right local units if and only if all its right ideals are unitary.
Proof. Straightforward.
Since UId(S) = USub( S S S ), the above results about biacts apply. So we know that UId(S) is a complete distributive lattice.
Recall that a quantale is a complete lattice L with an associative binary operation * (called multiplication) such that for all x, y k ∈ L, k ∈ K (where K is any index set). A quantale L is said to be unital if the semigroup (L, * ) is a monoid. Quantale homomorphisms are mappings between quantales that preserve multiplication and joins. A homomorphism of unital quantales has to preserve also the identity element.
The poset of unitary ideals of any semigroup has the structure of a quantale. where I = ∪ k∈K I k and J, I k , k ∈ K, are unitary ideals of S. Hence UId(S) is a quantale.

Remark 2.11.
A similar proof shows that also Id(S) is a quantale, where joins and product are defined in the same way as in UId(S), but meet is intersection.
Although UId(S) is a subset of Id(S), it is not a sublattice in general, because meet is computed in different ways in these lattices. However, UId(S) is a subquantale of Id(S).
Before formulating the next result note that every semigroup S has a largest factorisable subsemigroup, which we denote by U(S). It is possible to prove that U(S) is actually an ideal of S [6]. Proof. Necessity. Suppose that UId(S) has an identity element I with respect to multiplication. Then II = I, meaning that I is a factorisable subsemigroup of S. Therefore I lies in the largest factorisable subsemigroup U(S) of S. Since U(S) is factorisable and an ideal, we have U(S) ∈ UId(S). Additionally, for all J ∈ UId(S) we have whence J = U(S)J, and similarly J = JU(S). Thus U(S) is the identity element of the semigroup UId(S) and we conclude that I = U(S).
Sufficiency is clear.
A unital quantale is said to be integral if the multiplicative identity is the top element of the underlying lattice.

Unitary ideals and Morita contexts
Recall the following result about ideal lattices. Proposition 3.1 [4]. If S and T are Morita equivalent semigroups with weak local units then their ideal lattices are isomorphic.
Combining this result with Proposition 2.2 from [9] one can even say that the quantales of the ideals of such S and T are isomorphic.
The following example shows that Proposition 3.1 does not generalise to the class of firm semigroups. This is a factorisable semigroup with a commutative factorisable subsemigroup S = {0, 2, 3}. Since 2 is a right identity element of T and S, it follows immediately that these two semigroups are firm. We will show that T is an enlargement of S. This implies that they are strongly Morita equivalent as shown in the beginning of Section 2 in [7]. Since 0 = 0 · 0 · 0, 1 = 1 · 2 · 2, 2 = 2 · 2 · 2, Our next aim is to show that under relatively weak assumptions two semigroups will have isomorphic quantales of unitary ideals. For this we will need the following lemma.

Lemma 3.3. If Γ = (S, T, S P T , T Q S , θ, φ) is an acceptable Morita context and B T is a unitary T -act then
Thus B ⊆ B im(φ). The opposite inclusion is obvious. Now we can prove our main result.

Theorem 3.4. If two semigroups are connected by an acceptable Morita context then there exists an isomorphism between the quantales of their unitary ideals. This isomorphism takes finitely generated ideals to finitely generated ideals and principal ideals to principal ideals. If the quantales are unital then this isomorphism is an isomorphism of unital quantales.
Proof. 1. Let Γ = (S, T, S P T , T Q S , θ, φ) be an acceptable Morita context connecting semigroups S and T . For any unitary ideal J of T we see that the set θ(P J ⊗ Q) = {θ(pj ⊗ q) | p ∈ P, j ∈ J, q ∈ Q} is an ideal of S because θ is a homomorphism of (S, S)-biacts. We will prove that it is a unitary ideal. Using Lemma 3.3 we can calculate Then, precisely as in the proof of Theorem 3 in [4], one can show that Θ and Φ are inverses of each other. They also preserve order and hence they are isomorphisms of complete lattices. Moreover, for arbitrary I 1 , I 2 ∈ UId(S) we have where im(θ)I 2 = I 2 by the dual of Lemma 3.3. Similarly, Θ(J 1 )Θ(J 2 ) = Θ(J 1 J 2 ) for arbitrary J 1 , J 2 ∈ UId(T ). Thus Φ and Θ are isomorphisms of quantales. 2. We prove the claim about finitely generated ideals. Obviously, Φ(∅) = ∅ and Θ(∅) = ∅.
Take a unitary finitely generated ideal I = n j=1 S 1 a j S 1 of S and i ∈ {1, . . . , n}. AS a i ∈ I and I = I im(θ), we can write a i = s ki a ki θ(p ki ⊗ q ki ) for some k i ∈ {1, . . . , n}, s ki ∈ S 1 , p ki ∈ P and q ki ∈ Q. From I = im(θ)I it follows that a ki = θ(p li ⊗ q li )a li z li for some l i ∈ {1, . . . , n}, z li ∈ S 1 , p li ∈ P and q li ∈ Q. Hence Now, for any p ∈ P and q ∈ Q we have The opposite inclusion being clear, we conclude that 3. The relation above for n = 1 tells us that Φ maps unitary principal ideals to unitary principal ideals.
4. If UId(S) and UId(T ) are unital quantales then by Proposition 2.12 their identity elements are U(S) and U(T ). Note that U(S) is an idempotent in the quantale UId(S) such that I ⊆ U(S) for every other idempotent I. A similar statement is true for U(T ). Since the quantale isomorphisms preserve idempotents and order, it follows that Φ(U(S)) = U(T ) and Θ(U(T )) = U(S). So Φ and Θ are isomorphisms of unital quantales.
Hence UId(S) and UId(S n ) are isomorphic quantales. It follows that if S is an inflation of its subsemigroup T then UId(S) is isomorphic to UId(T ) because S 2 = T 2 .
We also have the following analogue of Theorem 2.5 in [9]. Theorem 3.9. If two semigroups S and T are connected by an acceptable Morita context (S, T, S P T , T Q S , θ, φ) then the following lattices are isomorphic: (1) UId(S), Proof. We only prove the isomorphism of (1) and (3) (the isomorphism of (2) and (4) is similar and the isomorphism of (1) and (2) is proved in Theorem 3.4).
Observe that if I ∈ UId(S) then IP is a sub-biact of S P T . We need to show that it is unitary. Since I = SI, we have IP = S(IP ). The inclusion IP T ⊆ IP is clear. We have to prove the inclusion IP ⊆ IP T . Take an element i 0 p 0 ∈ IP . Using the equality I = IS we can find sequences (i k ) k∈N ∈ I N and (s k ) k∈N ∈ S N such that i k = i k+1 s k+1 for every k ≥ 0. Since the context is acceptable, there exist n ∈ N, p ∈ P , q ∈ Q such that s n . . . s 2 s 1 = θ(p ⊗ q). Now Thus the equality IP = (IP )T also holds and IP ∈ USub( S P T ).
If A ∈ USub( S P T ) then using Lemma 3.3 we obtain An ideal I of a semigroup S is said to be factorisable if it is factorisable as a semigroup (that is, the equality I 2 = I holds).
Every factorisable ideal is unitary. The converse does not hold.

Example 3.10 (A unitary ideal need not be factorisable).
Let X + and X * be the free semigroup and the free monoid on an alphabet X, respectively. Then X + is a unitary non-factorisable ideal of X * .
The empty ideal is factorisable and any union of factorisable ideals of a given semigroup is a factorisable ideal. Hence the set of factorisable ideals is a complete lattice. Proof. By our assumptions I = Θ(J) = θ(P J ⊗ Q). Since, in particular, Θ is a semigroup homomorphism, one of I and J is factorisable precisely when the other is.
Let now I and J be factorisable ideals. It is easy to see that Γ = (I, J, IP J, JQI, θ , φ ), where the biact homomorphisms θ and φ are defined by  Proof. By Proposition 3.11 the mappings Θ and Φ restrict to mutually inverse isomorphisms between the lattices of factorisable ideals. Theorem 3.4 does not hold for one-sided ideals.
Example 3.13 (The lattices of (unitary) right ideals of Morita equivalent semigroups with local units need not be isomorphic). If S = {1} and T is a nonsingleton rectangular band then they have non-isomorphic lattices of (unitary) right ideals, although S is a monoid and T has local units. These semigroups are strongly Morita equivalent by Theorem 16 in [2].
One may also consider the posets of firm ideals. An ideal I of a semigroup S is said to be firm if it is firm as a right and left S-act. The following example shows that, contrary to Corollary 3.5, the posets of firm ideals of strongly Morita equivalent semigroups need not be isomorphic. Example 3.14. Consider the factorisable semigroup S = {0, 1, 2, 3} from Example 2.7 and its commutative subsemigroup T = {0, 2, 3}. This semigroup has a left identity element 3 which also belongs to T . This easily implies that S is an enlargement of T , and therefore S and T are (strongly) Morita equivalent.
The semigroup T has firm ideals ∅, {0}, {0, 2}, T and the semigroup S has firm ideals ∅, {0}, S. Thus the posets of firm ideals of S and T are not isomorphic.
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