# Acceptable Morita Contexts for Semigroups

Authors: Valdis Laan
Journal: ISRN Algebra
 (2012)

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## Abstract

This short note deals with Morita equivalence of (arbitrary) semigroups. We give a necessary and sufficient condition for a Morita context containing two semigroupsS and T to induce an equivalence between the category of closed right S-acts and the category of closed right T-acts.


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## Body

## 1. Preliminaries

Morita equivalence of two semigroups is usually defined by requiring that certain categories of acts over these semigroups are equivalent. If the semigroups are sufficiently “good” (in particular, if they are monoids) then the equivalence of those categories is equivalent to the existence of a unitary Morita context with surjective mappings (see, e.g., Theorems  7.3 and 8.3 of [1] or Theorem  1.1 of [2]). It is known (see Proposition  1 of [3]) that if two semigroups are contained in a unitary Morita context with surjective mappings (in such case they are called strongly Morita equivalent [4]), then these semigroups have to be factorisable, meaning that each of their elements can be written as a product of two elements. This suggests that perhaps the class of factorisable semigroups is the largest one to admit a meaningful Morita theory, and that theory is developed in [5]. Still in ring theory there are some articles that go beyond that limit and consider Morita equivalence of arbitrary associative rings. Our inspiration is taken from [6], where a connection between the equivalence of certain module categories over two associative rings and the existence of a certain Morita context is established. We shall prove an analogue of that result for semigroups.LetS be a semigroup. A leftS-act is a set A equipped with a left S-action S×A→A,(s,a)↦sa, such that s(s′a)=(ss′)a  for all s,s′∈S and a∈A. We write SA if A is a left S-act. Similarly right acts over semigroups are defined. If S and T are semigroups then an (S,T)-biact  SAT is a set A which is both left S-act and right T-act and (sa)t=s(at) for all s∈S, t∈T, a∈A. Clearly S is an (S,S)-biact with respect to actions defined by multiplication in S. A leftS-act morphism    ψ:AS→BS is a mapping ψ:A→B such that ψ(sa)=sψ(a) for all s∈S and a∈A. Similarly morphisms of right acts are defined. A biact morphism has to preserve both actions. Left S-acts (right T-acts, (S,T)-biacts) and their morphisms form a category.The tensor product  A⊗TB of acts AT and TB is the quotient set (A×B)/σ by the equivalence relation σ generated by the set {((at,b),(a,tb))∣a∈A,b∈B,t∈T}. The σ-class of (a,b)∈A×B is denoted by a⊗b. If SAT is an (S,T)-biact and TBS is a (T,S)-biact then the tensor product A⊗TB can be turned into an (S,T)-biact by setting s(a⊗b):=(sa)⊗b and (a⊗b)t:=a⊗(bt), a∈A, b∈B, s∈S, t∈T.A leftS-act SA over a semigroup S is unitary if SA=A. A biact SAT over semigroups S and T is unitary if SA=A and AT=A.Definition 1.1.
AMorita context is a sextuple (S,T,PTS,QST,θ,ϕ), where S and T are semigroups, SPT and TQS are biacts, and
(1.1)θ:S(P⊗TQ)S⟶SSS,ϕ:T(Q⊗SP)T⟶TTT,
are biact morphisms such that, for every p,p′∈P and q,q′∈Q,
(1.2)θ(p⊗q)p′=pϕ(q⊗p′),qθ(p⊗q′)=ϕ(q⊗p)q′.
Such a Morita context is called unitary if SPT and TQS are unitary biacts.


## 2. The Result

LetS be a semigroup. We consider the category FActS of unitary closed right S-acts AS  that is acts for which the canonical right S-act morphism μA:A⊗SS→A, defined by
(2.1)μA(a⊗s):=as,a∈A, s∈S, is an isomorphism. Since for a unitary act AS the mapping μA is obvioulsy surjective, the closedness of such AS is equivalent to injectivity of μA.Definition 2.1 (cf. Definition  3.6 of [6]).
We say that a Morita context(S,T,SPT,TQS,θ,ϕ) is right acceptable if  (1)
for every sequence(sm)m∈ℕ∈Sℕ there exists m0∈ℕ such that sm0⋯s1∈imθ,(2)
for every sequence(tn)n∈ℕ∈Tℕ there exists n0∈ℕ such that tn0⋯t1∈imϕ.Lemma 2.2.
LetS and T be factorisable semigroups. Then a Morita context (S,T,SPT,TQS,θ,ϕ) is right acceptable if and only if θ and ϕ are surjective.Proof. Necessity.
Let(S,T,SPT,TQS,θ,ϕ) be a right acceptable Morita context. Suppose that θ is not surjective. Then there is s∈S∖imθ. We may factorise it as s=s1′s1. We also may find for every i∈ℕ elements si+1′,si+1∈S such that si′=si+1′si+1. In particular, s=sk′sksk-1⋯s1 for every k∈ℕ. By right acceptability, for the sequence (sm)m∈ℕ there exists m0∈ℕ such that sm0⋯s1=θ(p⊗q) for some p∈P and q∈Q. But then
(2.2)s=sm0′sm0⋯s1=s′m0θ(p⊗q)=θ(s′m0p⊗q)∈imθ,
a contradiction. Similarly we obtain a contradiction if ϕ is not surjective.
Sufficiency is clear.Definition 2.3 (cf. [7], page 289).
We call a Morita context(S,T,SPT,TQS,θ,ϕ)rightFAct-pure if for every AS∈FActS and BT∈FActT the mappings
(2.3)1A⊗θ:A⊗P⊗Q⟶A⊗S,a⊗p⊗q⟼a⊗θ(p⊗q),1B⊗ϕ:B⊗Q⊗P⟶B⊗T,b⊗q⊗p⟼b⊗ϕ(q⊗p),
are injective.
Next we show how to construct closed acts over semigroups.Construction 1.
LetS be a semigroup and (sm)m∈ℕ∈Sℕ. We consider the free right S-act
(2.4)FS:=∐NS=⋃n∈N({n}×S),
with the right S-action
(2.5)(n,s)z:=(n,sz),
and its quotient act
(2.6)MS:=F/~={[k,s]∣k∈N,s∈S},
where the right S-act congruence ~ on F is defined by
(2.7)(k,s)~(l,z)⟺(∃n∈N)(n≥k,l∧sn⋯sk+1s=sn⋯sl+1z),k,l∈ℕ, s,z∈S, and the equivalence class of a pair (k,s) by ~ is denoted by [k,s]. Take [k,s]∈M, where k∈ℕ, s∈S. Since (sk+2sk+1)s=sk+2(sk+1s), we have (k,s)~(k+1,sk+1s), and hence
(2.8)[k,s]=[k+1,sk+1s]=[k+1,sk+1]s∈MS.
Consequently, MS=M and MS is unitary.
To prove the injectivity of the mappingμM:M⊗SS→M, suppose that μM([k,s]⊗u)=μM([l,z]⊗v) where k,l∈ℕ, s,z,u,v∈S. Then [k,su]=[l,zv] and hence sn⋯sk+1su=sn⋯sl+1zv for some n≥k,l. Consequently,
(2.9)[k,s]⊗u=[n+1,sn+1sn⋯sk+1s]⊗u=[n+1,sn+1]sn⋯sk+1s⊗u=[n+1,sn+1]⊗sn⋯sk+1su=[n+1,sn+1]⊗sn⋯sl+1zv=[n+1,sn+1]sn⋯sl+1z⊗v=[n+1,sn+1sn⋯sl+1z]⊗v=[l,z]⊗v,
in M⊗SS, so μM is an isomorphism and MS∈FActS.
Given a Morita context(S,T,SPT,TQS,θ,ϕ), one can always consider natural transformations γ:-⊗SP⊗TQ⇒1FActS and δ:-⊗TQ⊗SP⇒1FActT which are defined by
(2.10)γA:=μA∘(1A⊗θ):A⊗SP⊗TQS⟶ASδB:=μB∘(1B⊗ϕ):B⊗TQ⊗SPT⟶BT,AS∈FActS, BT∈FActT. Explicitly:
(2.11)γA(a⊗p⊗q)=aθ(p⊗q),δB(b⊗q⊗p)=bϕ(q⊗p),
where a∈A, b∈B, p∈P, q∈Q. We say that the natural transformations γ and δ are induced by the context. The following is an analogue of Theorem  3.10 of [6].Theorem 2.4.
Let(S,T,SPT,TQS,θ,ϕ) be a Morita context. The following assertions are equivalent.  (1)
FActS→-⊗sP←-⊗TQFActT are inverse equivalence functors with the natural isomorphisms induced by the context.(2)
The context(S,T,SPT,TQS,θ,ϕ) is right acceptable and right FAct-pure.Proof.
(
1
)
⇒
(
2
). If γA,δB are isomorphisms then clearly 1A⊗θ and 1B⊗ϕ have to be injective, so the context is right FAct-pure.
Consider now a sequence(sm)m∈ℕ∈Sℕ and MS as in Construction 1. Using surjectivity of γM:M⊗P⊗Q→M we can find p∈P, q∈Q, k∈ℕ, s∈S such that
(2.12)[1,s1]=γM([k,s]⊗p⊗q)=[k,s]θ(p⊗q)=[k,sθ(p⊗q)].
Hence
(2.13)sn⋯s1=sn⋯sk+1sθ(p⊗q)=θ(sn⋯sk+1sp⊗q)∈imθ
for some n∈ℕ, n≥k.
(
2
)
⇒
(
1
). Obviously γA is injective if AS∈FActS and the context is right FAct-pure. To prove that γA is surjective, take a∈A. By unitarity of AS, there exist (am)m∈ℕ∈Aℕ, (sm)m∈ℕ∈Sℕ such that a=a1s1 and ak=ak+1sk+1 for every k∈ℕ. Then for the sequence (sm)m∈ℕ there exists n∈ℕ, p∈P, q∈Q such that sn⋯s1=θ(p⊗q)∈imθ. Hence
(2.14)a=a1s1=a2s2s1=⋯=ansnsn-1⋯s1=anθ(p⊗q)=γA(an⊗p⊗q).
For δB the proof is similar.Corollary 2.5.
If for semigroupsS and T there exists a right acceptable and right FAct-pure Morita context then the categories FActS and FActT are equivalent.This corollary may be considered as a generalisation of Theorem  3 of [4], which states that if factorisable semigroups S and T are strongly Morita equivalent then they are Morita equivalent.



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*Source: 725627-2012-07-01.xml*