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[email protected] E tmrStein Itamar C Futi eirusadthe and Semigroups -Fountain soitdwith associated k Abstract S E Ersansmgop n euto Margolis of result a and semigroups -Ehresmann ≃ 1 k C (where S S aifistecnrec odto,there condition, congruence the satisfies hsgvsauie eeaiainfor generalization unified a gives This . ∗ dd Israel hdod, ftesmltie hswsgeneralized was This semilattice. the of a euti o udmna ntestudy the in fundamental now is result S hs tutr smr rnprn.A transparent. more is structure whose k ne e odtos eprove we conditions, few a Under . sayuia omttv ig if ring) commutative unital any is zto oacaso ih restriction right of class a to ization rvdta h eiru algebra semigroup the that proved o resmgopi smrhct the to isomorphic is semigroup erse 7 bandasmlrrsl for result similar a obtained [7] n etgt h algebra the vestigate aaa monoid, Catalan , migroup k S E-Ehresmann semigroups [19, 20] (E-Ehresmann semigroups were introduced by Lawson in [11]). This result has led to several applications regarding semigroups of partial functions [18, 21, 22, 13] and recently also to the study of certain partition monoids [3]. We mention also that Wang [26] generalized the above results further to a certain class of right P -restriction, P -Ehresmann semigroups (for definitions of these notions see [10]) - but we do not follow this approach in this paper. A hint for another direction is given by the Catalan monoid. The Catalan monoid Cn contains the order-preserving (x ≤ y =⇒ f(x) ≤ f(y)) and order-increasing (x ≤ f(x)) functions f on an n-element set. It is known that the algebra of the Catalan monoid is isomorphic to a certain incidence algebra ([8, Theorem 5.5] and [24, Theorem 17.25]) but recently Margolis and Steinberg obtained a simple proof of this fact using the change of basis approach [14]. Their result is not implied by any of the above-mentioned results. The goal of this paper is to obtain a generalization for the theorem on right restriction E-Ehresmann semigroups that includes also the case of the Catalan monoid. The class of semigroups which provides the correct context for this task is the class of reduced E-Fountain semigroups - also introduced by Lawson [12] under the name reduced E-semiabundant semigroups (which also appears in the literature as DR-semigroups [25]). Given a subset of idempotents E of S we can define two equivalence relations LE and RE on S. We say that aLE b (aRE b) if a and b have the same set of right (respectively, left)e identitiese from E. The semigroupe S ise called reduced E-Fountain if every LE and RE -class contains a (unique) idempotent from E and ef = e ⇐⇒ fe = e for every e,f ∈ eE. If ine addition LE and RE are right and left congruences respectively then we can associate a certain category Ce= C(S)ewith the semigroup S (for full details see [11]). We remark that E-Ehresmann semigroups are precisely those reduced E-Fountain semigroups which satisfy the congruence condition and whose distinguished subset of idempotents E forms a subsemilattice (i.e., a commutative subsemigroup of idempotents). For a,b ∈ S we define a relation El by the rule that a El b if a = be for some idempotent e ∈ E. This is a generalization of the right restriction partial order defined on E-Ehresmann semigroups - but

El is not a partial order or even a preorder. We also define an identity we call the generalized right ample condition which is a weak form of the right ample condition studied in the theory of E-Fountain semigroups. This background is described in Section 2 and Section 3 of the paper. Let S be a reduced E-Fountain semigroup which satisfies the congruence condition and assume also that sets of the form {b ∈ S | b El a} are finite for every a ∈ S. Let k be a commutative unital ring. In Section 4 we define a “change of basis” k-module homomorphism ϕ between the semigroup algebra kS and kC - the algebra of the associated category C. We show that ϕ is a homomorphism of k-algebras if and only if the generalized right ample identity holds. We also obtain an isomorphism in case El is contained in a partial order. In Section 5 we specialize our results in two cases. First, we show that if E is a subband (i.e., a subsemigroup of idempotents) then El is a partial order. Moreover, the generalized right ample condition and the standard one coincide thus obtaining a simpler result. Next, we discuss in detail the case of the Catalan monoid and show that we retrieve

2 the isomorphism described in [14]. Acknowledgments: The author would like to thank Professor Victoria Gould for a helpful con- versation.

2 Preliminaries

Let S be a semigroup and let S1 = S ∪{1} be the monoid formed by adjoining a formal unit element. Recall that Green’s preorders ≤R, ≤L and ≤J are defined by:

1 1 a ≤R b ⇐⇒ aS ⊆ bS 1 1 a ≤L b ⇐⇒ S a ⊆ S b 1 1 1 1 a ≤J b ⇐⇒ S aS ⊆ S bS

The associated Green’s equivalence relations on S are denoted by R, L and J . It is well known that L (R) is a right congruence (respectively, left congruence). A semigroup S is called J - trivial if J is the identity relation (that is, a J b ⇐⇒ a = b). Similar definitions hold for R-trivial and L-trivial semigroups. We denote by E(S) the set of idempotents of S. We denote by ≤ the natural partial order on E(S) defined by

e ≤ f ⇐⇒ (ef = fe = e).

Other elementary semigroup theoretic notions can be found in [9]. Let E ⊆ E(S) be some subset of idempotents. For every a ∈ S we denote by aE the set of right identities of a from E:

aE = {e ∈ E | ae = a}

Dually Ea is the set of left identities from E. We define two equivalence relations LE and RE on S by e e

aLEb ⇐⇒ (aE = bE) e aREb ⇐⇒ (E a =E b). e

It is easy to see that L ⊆ LE and R ⊆ RE . Recall that a subsemigroup of idempotents E ⊆ S is called a subband. A commutativee subbande is called a subsemilattice. Let k be a commutative unital ring. The semigroup algebra kS of a semigroup S is defined in the following way. It is a free k-module with basis the elements of S, that is, it consists of all formal

3 linear combinations

{k1s1 + ... + knsn | ki ∈ k, si ∈ S}.

The multiplication in kS is the linear extension of the semigroup multiplication. We will also need the notion of a category algebra in this paper. The category algebra kC of a (small) category C is defined in the following way. It is a free k-module with basis the morphisms of C, that is, it consists of all formal linear combinations

1 {k1m1 + ... + knmn | ki ∈ k, mi ∈C }.

The multiplication in kC is the linear extension of the following:

′ ′ ′ m m if m · m is defined m · m =  0 otherwise.  Let R be a relation on a set X. We say that R is principally finite if the set {x′ ∈ X | x′Rx} is finite for every x ∈ X. If R = is a principally finite partial order we can define the incidence algebra of  which consists of all functions f :→ k with standard addition operation and the multiplication operation is defined by f⋆g(a,b)= f(a,c)g(c,b). X acb We denote this algebra by k[]. It is well known that an element f ∈ k[] is invertible if and only if f(x, x) is invertible in k for every x ∈ X. For other elementary facts on incidence algebras see [17, Section 3.6]). Another point is worth mentioning. The poset  can also be viewed as a category whose set of objects is X and there exists a unique morphism from an object x1 to another object x2 if and only if x1  x2. The category algebra in general differs from the incidence algebra because the elements of the category algebra are only finite linear combinations. However, the two notions coincide if X is a finite set.

3 Reduced E-Fountain Monoids

3.1 Basic definitions

Definition 3.1. A semigroup S is called E-Fountain if every LE-class contains an idempotent from

E and every RE-class contains an idempotent from E, where Ee ⊆ S is some subset of idempotents. We remark thate this property is also called “E-semiabundant” in the literature.

The following is an immediate consequence of [25, Propositions 1.2 and 1.3].

4 Proposition 3.2. Let S be an E-Fountain semigroup. The following conditions are equivalent:

1. ∀e,f ∈ E, ef = e ⇐⇒ fe = e.

2. For every a ∈ S the sets aE and Ea contain a minimum element (with respect to the natural partial order on idempotents).

3. We can equip S with two binary operations ∗ and + which satisfy the following identities for every a,b ∈ S:

a+a = a (a+)+ = a+ a+(ab)+ = (ab)+a+ = (ab)+ aa∗ = a (a∗)∗ = a∗ b∗(ab)∗ = (ab)∗b∗ = (ab)∗ (a+)∗ = a+, (a∗)+ = a∗

∗ + It is important to note that a (a ) is the minimum element of aE (respectively, Ea) and that e∗ = e+ = e for every e ∈ E. We follow [12] and call an E-Fountain semigroup reduced if it satisfies the equivalent conditions of Proposition 3.2. Such a semigroup is called a “DR-semigroup” in [25]. It is clear from Condition 3 that the class of all reduced E -Fountain semigroup is a variety of bi-unary semigroups.

Definition 3.3. Let S be a reduced E-Fountain semigroup. We say that S satisfies the congruence condition if LE is a right congruence and RE is a left congruence. e e It is well known that S satisfies the congruence condition if and only if the identities (ab)∗ = (a∗b)∗ and (ab)+ = (a+b)+ hold - see [5, Lemma 4.1]. In this case we can define a category C(S) in the following way. The objects are in one-to-one correspondence with the set E. The morphisms are in one to one correspondence with elements of S. The convention is to compose morphisms in Ehresmann categories “from left to right”. However, for us it will be more convenient to use composition “from right to left”. Therefore, for every a ∈ S the associated morphism C(a) has domain a∗ and range a+. If the range of C(a) is the domain of C(b) (that is, if b∗ = a+) the composition C(b) · C(a) is defined to be C(ba). The assumption b∗ = a+ implies that (ba)+ = (ba+)+ = (bb∗)+ = b+ and likewise (ba)∗ = a∗ so this is indeed a category - see [11] for additional details. We remark that if the set E is a subsemilattice then a reduced E-Fountain semigroup which satisfies the congruence condition is called an E-Ehresmann semigroup (see [5, 6]).

3.2 The relations ≤l and El

Let S be a reduced E-Fountain semigroup. The following partial order is defined in [11]:

∗ ∗ ∗ a ≤l b ⇐⇒ (a ≤ b and a = ba )

5 This is a generalization of the right “restriction” partial order of an E-Ehresmann semigroup and in particular of the natural partial order of an inverse semigroup. However, we will need a different generalization for our purpose.

Definition 3.4. Let a,b ∈ S. We define a El b ⇐⇒ a = be for some e ∈ E.

∗ Lemma 3.5. a El b ⇐⇒ a = ba .

Proof. The ⇐ direction is immediate. For the other direction assume a = be so ae = a hence a∗ ≤ e in E. Therefore a = aa∗ = a(ea∗) = (ae)a∗ = ba∗ as required.

The following uniqueness property will also be useful.

∗ ∗ Lemma 3.6. If a El b then a is the unique x ∈ S such that x El b and x = a .

∗ ∗ ∗ ∗ Proof. Assume x El b such that x = a . Then x = bx = ba = a.

It is clear that ≤l⊆El and it is easy to see that e ≤ f ⇐⇒ e ≤l f ⇐⇒ e El f for e,f ∈ E. It ∗ is tempting to think that El is also a partial order. It is clear that El is reflexive since a = aa . However, the following example shows that in general it is not antisymmetric and in Remark 5.13 we will give a counter-example for transitivity.

Example 3.7. [26, Example 2.3] Let S be a “square” rectangular band. The elements of S are pairs (i, j) where 1 ≤ i, j ≤ n for some fixed n ∈ N. Multiplication is defined by

(i1, j1) · (i2, j2) = (i1, j2).

Recall that (i1, j1) R(i2, j2) ⇐⇒ i1 = i2 and (i1, j1) L(i2, j2) ⇐⇒ j1 = j2. Every element of S is an idempotent, but we choose E to be the set of “diagonal” elements E = {(i,i) | 1 ≤ i ≤ n}. It is easy to see that (i, j)e = (i, j) ⇐⇒ e = (j, j) for e ∈ E. Therefore, the set of right identities of (i, j) contains a unique idempotent from E and the dual claim holds for left identities. Hence, S is a reduced E-Fountain semigroup. It is also easy to check that (i1, j1) El (i2, j2) if and only if i1 = i2 . Therefore, a El b if and only if a R b so El is in fact a symmetric relation! On the other hand, ≤ is the trivial relation on E so ≤l is also trivial on S. We give another simple observation for future use. It is easy to verify that (i1, j1)LE(i2, j2) if and only if j1 = j2. Therefore, LE = L so LE is a right congruence and and dually REe= R is a left congruence so S satisfies alsoe the congruencee condition. e

6 Clearly, we can also define a dual relation a Er b ⇐⇒ a = eb for some e ∈ E.

3.3 The right ample and the generalized right ample conditions

Let S be a reduced E-Fountain semigroup which satisfies the congruence condition. We say that the right ample condition (or right ample identity) holds in S if ea = a(ea)∗ for every a ∈ S and e ∈ E. This is equivalent to the condition Ea ⊆ aE for every a ∈ S. The right ample condition is well studied, but it is a too strong requirement for some of the monoids we want to consider.

Definition 3.8. Let S be a reduced E-fountain semigroup which satisfies the congruence condition. We say that the generalized right ample condition (or generalized right ample identity) holds in S if + ∗ + e a (eaf)∗ = a (eaf)∗ 

for every a ∈ S and e,f ∈ E.

This identity can also be written as

+ ∗ + b∗ a (b∗ac∗)∗ = a (b∗ac∗)∗ 

for every a,b,c ∈ S hence the class of reduced E-Fountain semigroups which satisfy the congruence condition and the generalized right ample identity is also a variety of bi-unary semigroups. In the rest of this section we show that if S satisfies the right ample condition it satisfies also the generalized right ample condition hence justifying our term “generalized”. We fix a reduced E-Fountain semigroup S which satisfies the congruence condition.

∗ ∗ Lemma 3.9. If the right ample condition holds in S then b El c =⇒ b ≤ c .

∗ Proof. If b El c then b = cb so b∗ = (cb∗)∗ = (c∗b∗)∗.

The right ample identity implies: c∗b∗ = b∗(c∗b∗)∗ and by the above equality b∗(c∗b∗)∗ = b∗b∗ = b∗.

Therefore c∗b∗ = b∗ hence b∗ ≤ c∗ as required.

We record the following immediate corollary.

Corollary 3.10. If the right ample condition holds in S then El=≤l.

7 Proposition 3.11. If the right ample condition holds in S then so is the generalized right ample condition.

∗ ∗ ∗ Proof. Let e,f ∈ E and a ∈ S. Since eaf El ea, Lemma 3.9 implies that (ea) (eaf) = (eaf) . This and the right ample identity implies that

ea(eaf)∗ = a(ea)∗(eaf)∗ = a(eaf)∗

∗ ∗ ∗ + ∗ + ∗ + so e is a left identity of a(eaf) . Since a(eaf) RE(a(eaf) ) we obtain e(a(eaf) ) = (a(eaf) ) as well. Finally, e + ∗ + ∗ + e a (eaf)∗ = a (eaf)∗ = a (eaf)∗ 
 

so S satisfies the generalized right ample condition.

Example 3.12. The rectangular band from Example 3.7 satisfies the generalized right ample + condition but not the right ample condition. If a (eaf)∗ = (i,i) and e = (k, k) then

+ ∗ + e a (eaf)∗ = ((k, k)(i,i))∗ = (k,i)∗ = (i,i)= a (eaf)∗ 

so the required equality holds for every e,f ∈ E, and a ∈ S. On the other hand, we have already seen that ≤l6=El so the right ample condition doesn’t hold.

Remark 3.13. We can of course define the dual notion. We say that S satisfies the generalized left ample condition if the identity

+ + ∗ + ∗ (fae) a e = (fae) a holds for every a ∈ S and e,f ∈ E.

4 The semigroup and category algebras

4.1 Homomorphism of algebras

In this section we fix a reduced E-Fountain semigroup S which satisfies the congruence condition.

We also assume that the relation El is principally finite. Let C be the associated category as defined in Section 3 and let k be a commutative unital ring. Define ϕ : kS → kC on basis elements by

ϕ(a)= C(c). X cEla

8 It is clear that ϕ is a k-module homomorphism. We want to show that it is an algebra homomor- phism if and only if the generalized right ample condition holds in S. For this we need a different formulation of this identity.

Lemma 4.1. The following conditions are equivalent.

1. The semigroup S satisfies the generalized right ample condition.

∗ + ∗ ∗ + 2. The implication c El ea =⇒ (e(ac ) ) = (ac ) holds for every a,c ∈ S and e ∈ E.

∗ + ∗ ∗ + 3. The implication c El ba =⇒ (b(ac ) ) = (ac ) holds for every a,b,c ∈ S.

Proof.

∗ + ∗ ∗ + (1 =⇒ 2) If c El ea then c = eaf for some f ∈ E. The equality (e(ac ) ) = (ac ) follows immediately from the generalized right ample identity.

∗ ∗ ∗ ∗ (2 =⇒ 3) Assume c El ba so c = bac . Denote d = b ac so clearly d El b a. Our assumption implies

(b∗(ad∗)+)∗ = (ad∗)+.

Now, d∗ = (b∗ac∗)∗ = (bac∗)∗ = c∗

so the above equality can be written

(b∗(ac∗)+)∗ = (ac∗)+.

Therefore, (b(ac∗)+)∗ = (b∗(ac∗)+)∗ = (ac∗)+

which finishes the proof.

(3 =⇒ 1) First substitute b = e for e ∈ E and then note that eaf El ea so we can choose c = eaf and the assumption implies + ∗ + e a (eaf)∗ = a (eaf)∗ 

as required.

Theorem 4.2. The module homomorphism ϕ is a homomorphism of k-algebras if and only if the generalized right ample identity holds in S.

9 Proof. We start with the “if” part. For a,b ∈ S, we need to show that ϕ(ba)= ϕ(b)ϕ(a). We need to prove that C(c)= C(c′′) C(c′). X X′′ X′ cElba c Elb c Ela

′′ ′ ′′ ′ ′′ ∗ ′ + First note that if c El b and c El a and C(c )C(c ) 6=0 then (c ) = (c ) and so

c′′c′ = b(c′′)∗c′ = b(c′)+c′ = bc′ = ba(c′)∗

′′ ′ so c c El ba. Therefore, every element on the right-hand side appears also on the left-hand side. ′ ∗ ′′ ∗ + Now, take c El ba and define c = ac and c = b(ac ) . From the generalized right ample condition we obtain (c′′)∗ = (b(ac∗)+)∗ = (ac∗)+ = (c′)+.

Therefore, the composition C(c′′) · C(c′) is defined. Now

c′′c′ = b(ac∗)+ac∗ = bac∗ = c so every element from the left-hand side appears on the right-hand side. It remains to show that ′′ ′ ′′ ′ ′ ∗ ∗ it appears only once. Assume C(c) = C(d )C(d ) for d El b and d El a. Then (d ) = c so by Lemma 3.6 we must have d′ = ac∗ = c′. Since C(d′′) · C(d′) is defined we must have (d′′)∗ = (d′)+ = (ac∗)+ so again Lemma 3.6 implies d′′ = b(ac∗)+ = c′′. This proves uniqueness. The “only if” part comes from another examination of the above argument. Let a,b ∈ S , if ϕ is a homomorphism of algebras we must have ϕ(ba)= ϕ(b)ϕ(a) so

C(c)= C(c′′) C(c′). X X′′ X′ cElba c Elb c Ela

Choose any c El ba from the left-hand side. The morphism C(c) must appear on the right-hand ′ ′′ ′′ ′ ′ ∗ ∗ side. So there exists c El a and c El b such that C(c) = C(c )C(c ). This implies (c ) = c so c′ = ac∗. Since the product C(c) = C(c′′)C(c′) is defined it must be the case that (c′′)∗ = (ac∗)+ and therefore c′′ = b(ac∗)+. This implies (b(ac∗)+)∗ = (ac∗)+ as required.

4.2 Isomorphism of algebras

In general, ϕ is not an isomorphism. For instance, consider Example 3.7. We have already seen (Example 3.12) that it satisfies the requirements of Theorem 4.2 so ϕ is an homomorphism of k-algebras. But in this example we have ϕ(a)= ϕ(b) if a R b so ϕ is not injective.

However, we can prove that ϕ is an isomorphism if El is contained in a (principally finite) partial order.

10 Lemma 4.3. Let S be a reduced E-Fountain semigroup which satisfies the congruence condition.

Assume El⊆ where  is some principally finite partial order. Then ϕ is an isomorphism of k-modules.

Proof. Consider the incidence algebra k[]. Define ζl ∈ k[] to be the zeta function of El:

1 a El b ζl(a,b)=  0 otherwise  −1 As ζl(a,a)=1 for every a ∈ A, we know that ζl has an inverse ζl . This means that

1 a = b ζ−1 ⋆ ζ (a,b)= ζ−1(a,c)ζ (c,b)= δ(a,b)=  l l X l l  acc 0 a 6= b  and likewise −1 ζl ⋆ ζl (a,b)= δ(a,b).

Note that ϕ(a)= C(b)= ζ (b,a)C(b). X X l bEla ba The inverse of ϕ is given by ψ(C(a)) = ζ−1(b,a)b. X l ba Indeed

ψ(ϕ(a)) = ψ( ζ (b,a)C(b)) X l ba = ζ (b,a)ψ(C(b)) X l ba

= ζ (b,a)  ζ−1(c,b)c X l X l ba cb  = c ζ−1(c,b)ζ (b,a) X X l l ca cba = cδ(c,a)= a X ca and a similar argument shows that ϕ(ψ(C(a)) = C(a).

For future reference we state clearly the following immediate corollary of Theorem 4.2 and Lemma 4.3.

11 Theorem 4.4. If El is contained in some (principally finite) partial order then ϕ is an isomorphism of k-algebras if and only if the generalized right ample condition holds in S.

We will see later that ϕ is indeed contained in a partial order for a few common cases.

5 Special subcases

We now specify the results of Section 4 for two subcases.

5.1 The case where E is a subband

In this section we fix S to be a reduced E-Fountain semigroup which satisfies the congruence condition where the subset S is a subband (i.e., a subsemigroup of idempotents).

Proposition 5.1. Under the above assumptions we have ≤l=El on S.

∗ Proof. We have already seen that ≤l⊆El. For the other direction assume a El b hence a = ba this implies that a∗ = (ba∗)∗ = (b∗a∗)∗.

The fact that E is a subband implies (b∗a∗)∗ = b∗a∗ so a∗ = b∗a∗ hence a∗ ≤ b∗. This implies ∗ ∗ a ≤l b as required.

In particular, this implies that if E is a subband the relation El is a partial order. The main goal in this section is to show that if E is a subband then the generalized and standard right ample identities are equivalent.

Proposition 5.2. Let S be a reduced E-Fountain semigroup which satisfies the congruence condi- tion where E is a subband. Then the right ample identity holds if and only if the generalized right ample identity holds.

Proof. In view of Proposition 3.11, it is enough to show that the generalized right ample identity implies the standard one. Let a ∈ S and e ∈ E. If E is a subband then for every f ∈ E we have + e a (eaf)∗ ∈ E and therefore

+ ∗ + e a (eaf)∗ = e a (eaf)∗ . 


The assumption that S satisfies the generalized right ample identity implies

+ + e a (eaf)∗ = a (eaf)∗

12 ∗ + ∗ ∗ + so e is left identity of a (eaf) . Since a(eaf) RE (a(eaf) ) this is equivalent to
e ea (eaf)∗ = a (eaf)∗

Now we can substitute f = a∗ and obtain

ea = ea(ea)∗ = ea(eaa∗)∗ = a (eaa∗)∗ = a (ea)∗ as required.

As an immediate corollary of Theorem 4.4, Proposition 5.2, and Proposition 5.1 we obtain:

Corollary 5.3. Let S be a reduced E-Fountain semigroup which satisfies the congruence condition where E ⊆ S is a subband and El=≤l is principally finite. Then ϕ is an isomorphism of k-algebras if and only if the right ample identity holds in S.

Remark 5.4. Consider the more specific case where E is a subsemilattice of S so S is an E- Ehresmann semigroup. In this case Corollary 5.3 is already known. The fact that ϕ is an iso- morphism of k-algebras was proved by the author in [19, 20]. The necessity of the right restriction property was proved by Wang in [26, Lemma 4.3]. In fact, Wang proved a version of Corollary 5.3 to the class of right P -restriction locally Ehresmann P -Ehresmann semigroups (for definitions of these notions, see [10, 26]). We leave open the problem of finding a unified generalization for Theorem 4.2 and Wang’s result [26, Theorem 4.4].

5.2 Catalan monoid

A function f :[n] → [n] (where [n]= {1,...,n}) is called order-preserving if i ≤ j =⇒ f(i) ≤ f(j) for every i, j ∈ [n] and order-increasing if i ≤ f(i) for every i ∈ [n]. Denote by Cn the monoid of all order-preserving and order-increasing functions f :[n] → [n], called the Catalan monoid. The name comes from the fact that its size is the n-th Catalan number [4, Theorem 14.2.8]. The Catalan monoid is J -trivial [24, Proposition 17.17] and it is well known that every J -trivial monoid is E-Fountain for E = E(S) [15, Corollary 3.2]. It is also known that in a J -trivial monoid ef = e ⇐⇒ fe = e ⇐⇒ e≤J f for e,f ∈ E(S) ([2, Lemma 3.6]) so every J -trivial monoid is in fact a reduced E-Fountain monoid. We define a partial order n on subsets of [n] in the following way. For two subsets X = {x1 <...

(i.e., |X| = |Y |) and xi ≤ yi for every 1 ≤ i ≤ k. It is well known (see [14]) that elements of Cn+1 are in one-to-one correspondence with pairs (X, Y ) where X, Y ⊆ [n] and X n Y . In other words, there is a one-to-one correspondence between elements of Cn+1 and elements of n viewed as a set

13 of ordered pairs. Explicitly, given such a pair (X, Y ) we define

y1 if 1 ≤ i ≤ x1  fX,Y (i)= yj if xj−1

It is easy to verify that fX,Y ∈ Cn+1. Note that im(f)\{n +1} = Y and that

ker(fX1,Y1 ) = ker(fX2,Y2 ) ⇐⇒ X1 = X2

(where im(f) is the image of f and ker(f) is the equivalence relation on [n] defined by (x1, x2) ∈ ker f if and only if f(x1)= f(x2)). Note also that xi is the maximal element whose image is yi. Recall that f :[n] → [n] is an idempotent if and only if f(i) = i for every i ∈ im(f). Therefore fX,Y is an idempotent if and only if X = Y . In particular, there is a one-to-one correspondence between idempotents in Cn+1 and subsets of [n]. For every Z ⊆ [n] we denote by eZ the idempotent corresponding to Z. Explicitly (see [24, Proposition 17.18]):

eZ (i) = min{z ∈ Z ∪{n +1} | i ≤ z}

The following proposition is part of [24, Proposition 17.20] (with notation f − instead of our f ∗).

∗ + Proposition 5.5. Let fX,Y ∈ Cn+1 then (fX,Y ) = eX and (fX,Y ) = eY .

As an immediate corollary we have:

Corollary 5.6. Let fX1,Y1 ,fX2,Y2 ∈ Cn+1 then

fX1,Y1 LEfX2,Y2 ⇐⇒ X1 = X2 ⇐⇒ ker(fX1,Y1 ) = ker(fX2,Y2 ) e im im fX1,Y1 REfX2,Y2 ⇐⇒ Y1 = Y2 ⇐⇒ (fX1,Y1 )= (fX2,Y2 ). e Lemma 5.7. The Catalan monoid satisfies the congruence condition.

Proof. Take two functions f,g ∈ Cn+1. For LE to be a right congruence we need to show that (fg)∗ = (f ∗g)∗ or equivalently, ker(fg) = ker(fe∗g). Indeed,

(a1,a2) ∈ ker(fg) ⇐⇒ fg(a1)= fg(a2) ⇐⇒ (g(a1),g(a2)) ∈ ker f ∗ ∗ ∗ ⇐⇒ (g(a1),g(a2)) ∈ ker f ⇐⇒ f g(a1)= f g(a2) ∗ ⇐⇒ (a1,a2) ∈ ker(f g).

14 + + + To show that RE is a left congruence we need to show that (fg) = (fg ) or equivalently, im(fg)= im(fge+). Indeed,

b ∈ im(fg) ⇐⇒ ∃a ∈ im(g), f(a)= b ⇐⇒ ∃a ∈ im(g+), f(a)= b ⇐⇒ b ∈ im(fg+).

Note that the objects of the associated category are subsets of [n] and if X, Y ⊆ [n] then there is a unique morphism from X to Y if and only if X n Y . Therefore the associated category is just the poset n viewed as a category. Our next goal is to prove that the generalized right ample identity holds in the Catalan monoid. The proof given here is essentially taken from [14], but we reformulate it in the language of E-Fountain monoids. We first need the following definition.

Definition 5.8. Let f ∈ Cn+1 and let Z ⊆ [n], we say that Z is a partial cross section (PCS) of f if n +1 ∈/ f(Z) and f|Z is injective (where f|Z is the restriction of f to the set Z).

∗ Proposition 5.9. Let f ∈ Cn+1 and let Z ⊆ [n] then (feZ ) = eZ if and only if Z is a PCS of f.

∗ Proof. First assume (feZ ) = eZ . Let z ∈ Z. Since eZ and feZ have the same kernel and ′ eZ (z) = z 6= n +1= eZ (n + 1) then f(z) = feZ (z) 6= feZ(n +1) = n +1. Now if f(z) = f(z ) ′ ′ ′ ′ for some z,z ∈ Z then feZ (z)= feZ(z ) hence (z,z ) ∈ ker(feZ ). Therefore, (z,z ) ∈ ker(eZ ) so ′ ′ z = eZ (z)= eZ (z ) = z and f|Z is indeed injective. In the other direction assume Z is a PCS of f. We need to prove that ker(feZ ) = ker(eZ ). It is clear that ker(eZ ) ⊆ ker(feZ ). For the other ′ ′ containment assume (x, x ) ∈ ker(feZ ) so feZ(x)= feZ (x ). First note that

′ ′ feZ (x)= feZ (x )= n +1 ⇐⇒ eZ (x)= eZ (x )= n +1

′ because im(eZ ) = Z ∪{n +1} and n +1 ∈/ f(Z). The other option is that eZ (x),eZ (x ) ∈ Z. In this case the fact that f|Z is injective implies that

′ ′ feZ (x)= feZ(x ) =⇒ eZ (x)= eZ (x ) so ker(feZ ) ⊆ ker(eZ ) as required.

∗ Corollary 5.10. If h El f for h,f ∈ Cn+1 and h = eZ where Z ⊆ [n] then Z is a PCS of f.

∗ ∗ Proof. We have h = feZ so eZ = h = (feZ ) and the result follows by Proposition 5.9.

Theorem 5.11. The Catalan monoid Cn+1 satisfies the generalized right ample condition.

15 Proof. We use the equivalent condition 3 given in Lemma 4.1. Assume h El fg for some f,g,h ∈ Cn+1 ∗ and denote h = eZ . Since h El fg we know that Z is a PCS of fg by Corollary 5.10. Denote

W = im(geZ )\{n +1} and note that for every w ∈ W there exists z ∈ Z such that g(z)= geZ (z)= w. We need to prove that + ∗ + (f(geZ) ) = (geZ ) .

According to Proposition 5.9 and Proposition 5.5 we need to show that W is a PCS of f. Take w ∈ W and let z ∈ Z such that g(z)= w, then f(w)= fg(z) 6= n +1 since Z is a PCS of fg. Now take w, w′ ∈ W such that f(w)= f(w′). Take z,z′ ∈ Z such that g(z)= w and g(z′)= w′. Then ′ ′ fg(z)= fg(z ) so z = z since fg|Z is injective.

Let S be a reduced E-Fountain semigroup. It is known [1, Lemma 2.3] that if E = E(S) and the right ample identity holds then E is a subband. Therefore, the Catalan monoid Cn (for n ≥ 3) is another example of a semigroup which satisfies the generalized right ample condition but not the standard one. A concrete simple counter-example is given in the following example:

Example 5.12. Consider the idempotents e1,e2 ∈ C3 defined by

2 i =1, 2 1 i =1 e1(i)=  , e2(i)=  3 i =3 3 i =2, 3   so 2 i =1 e1e2(i)=  3 i =2, 3.  ∗ It is easy to see that (e1e2) = e2 as ker e2 = ker e1e2. Therefore,

∗ e2(e1e2) = e2e2 = e2 6= e1e2 so the right ample condition doesn’t hold.

Remark 5.13. Example 5.12 also shows that the relation El in Cn is not transitive and hence not even a preorder. It is clear that e1 El id (where id is the identity function) and e1e2 El e1. However, e1e2 is not an idempotent hence e1e2 5l id.

Fortunately, it is easy to see that El is contained in a partial order.

Lemma 5.14. Let S be a reduced E-Fountain semigroup. If S is R-trivial then El is contained in a partial order.

∗ Proof. If a El b then a = ba so a ≤R b hence El⊆≤R. The relation ≤R is a partial order if and only if S is R-trivial.

16 In particular, ≤R is a partial order in the Catalan monoid which is a J -trivial monoid. In conclusion, the results in this section and Theorem 4.4 implies the following corollary:

Theorem 5.15. Let k be a unital commutative ring. There is an isomorphism of algebras k Cn+1 ≃ k[n].

In particular, k Cn+1 is an incidence algebra.

Remark 5.16. This isomorphism is precisely the one given in [14, Theorem 3.1]. As a final observation, we consider also the generalized left ample condition.

Proposition 5.17. The Catalan monoid Cn+1 satisfies the generalized left ample condition.

+ Proof. Note that (eZ g) = eZ means that eZ g and eZ have the same image. In other words, for every z ∈ Z there exists x such that eZ g(x)= z. In this case we say that g is a multi cross section (MCS) of Z. We use the dual of condition 3 given in Lemma 4.1:

+ ∗ + + ∗ h Er fg =⇒ ((h f) g) = (h f)

+ + + + + If h Er fg and h = eZ then eZ = h = (h fg) = (eZ fg) so fg is an MCS of Z. Denote ∗ eW = (eZ f) . We need to prove that g is an MCS of W . Let w ∈ W , we need to find x such that eW g(x)= w = eW (w). Note that eW and eZ f have the same kernel so this is equivalent to finding x such that eZ fg(x)= eZ f(w). Set z = eZ f(w) so z ∈ im(eZ ). If z = n+1 then eZ f(w)= eZ f(n+1) so w = eW (w)= eW (n +1) = n +1, a contradiction. Therefore, z ∈ im(eZ )\{n +1} = Z. Since fg is an MCS of Z there exists an x such that eZ fg(x)= z as required.

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