arXiv:1702.00059v1 [math.GR] 31 Jan 2017
PARTIAL ACTIONS AND A THEOREM BY O’CARROLL MYKOLA KHRYPCHENKO Abstract. We give a simple proof of a theorem by L. O’Carroll, which uses partial actions of inverse semigroups.
Introduction It was proved by D. McAlsiter in [7, Theorem 2.6] that, up to an isomorphism, each E-unitary inverse semigroup is of the form P (G, X, Y ) for some McAlister triple (G, X, Y ). There are many alternative proofs of this fact (see, for example, the survey [5]), in particular, the one given in [4] uses partial actions of groups on semilattices. L. O’Carroll generalized in [8, Theorem 4] the McAlister’s P -theorem by showing that, given an inverse semigroup S and an idempotent pure congruence ρ on S, there is a fully strict L-triple (T, X, Y ) with T ∼ = S/ρ and Y ∼ = E(S), such that ∼ S = Lm (T, X, Y ). When S is E-unitary and ρ is the minimum group congruence on S, this coincides with the McAlister’s description of S as P (G, X, Y ). In this short note we use partial actions of inverse semigroups on semilattices to get a simple proof of [8, Theorem 4]. 1. Preliminaries Given an inverse semigroup S, we use the standard notations E(S) for the semilattice of idempotents of S and ≤ for the natural partial order on S. A congruence ρ on S is said to be idempotent pure, if (e, s) ∈ ρ with e ∈ E(S) implies s ∈ E(S). The semigroup S is called E-unitary, if its minimum group congruence σ, defined by (s, t) ∈ σ ⇔ ∃u ≤ s, t, is idempotent pure. An F -inverse monoid is an inverse semigroup, in which every σ-class has the maximum element under ≤. Each F -inverse monoid is E-unitary. Let S and T be inverse semigroups. Recall from [4] that a map τ : S → T is called a premorphism1, if (i) τ (s−1 ) = τ (s)−1 ; (ii) τ (s)τ (t) ≤ τ (st). Observe from (i) that τ (e) = τ (e)−1 for any e ∈ E(S). Moreover, by (ii) one has τ (e)τ (e)−1 = τ (e)τ (e) ≤ τ (e), whence τ (e)τ (e)−1 = τ (e)τ (e)−1 τ (e) = τ (e). So, τ (E(S)) ⊆ E(T ). 2010 Mathematics Subject Classification. Primary 20M18, 20M30; Secondary 20M15. Key words and phrases. Inverse semigroup, premorphism, idempotent pure congruence. Supported by FAPESP of Brazil (process number: 2012/01554–7). 1In [6, 9] such a map is called a dual prehomomorphism. 1
2
MYKOLA KHRYPCHENKO
Given a set X, by I(X) we denote the symmetric inverse monoid on X. The elements of I(X) are partial bijections of X, where the product of f, g ∈ I(X) is f ◦ g : g −1 (dom f ∩ ran g) → f (dom f ∩ ran g). By a partial action of an inverse semigroup S on a set X we mean a premorphism τ : S → I(X). We use the notation τs for the partial bijection τ (s). 2. Construction Lemma 2.1. Let S be an inverse semigroup, X a set, ρ an idempotent pure congruence on S and τ a partial action of S on X. If (s, t) ∈ ρ and x ∈ dom τs ∩dom τt , then τs (x) = τt (x). Proof. It follows from (s, t) ∈ ρ that (st−1 , tt−1 ) ∈ ρ, so st−1 ∈ E(S), as ρ is idempotent pure. Therefore, τst−1 , being an idempotent of I(X), is the identity −1 ≤ τst−1 by (ii), we have map on its domain. Since τ −1 t−1 = τt by (i) and τs ◦ τt −1 −1 −1 τt (dom τs ∩ dom τt ) = τ −1 t−1 (dom τs ∩ ran τt ) = dom (τs ◦ τt ) ⊆ dom τst .
Hence, τt (x) ∈ dom τst−1 and τst−1 ◦ τt (x) = τt (x).
(1)
On the other hand, iddom τt = τ −1 t ◦ τt ≤ τt−1 t , so x ∈ dom τt−1 t , τt−1 t (x) = x ∈ dom τs and τs ◦ τt−1 t (x) = τs (x).
(2)
It remains to note that both τst−1 ◦ τt and τs ◦ τt−1 t are restrictions of τst−1 t thanks to (ii), whence τs (x) = τt (x) by (1) and (2). Definition 2.2. Under the conditions of Lemma 2.1 for any class [s] ∈ S/ρ define [ [ ran τt dom τt → τ˜[s] : t∈[s]
t∈[s]
by τ˜[s] (x) = τt (x), if t ∈ [s] and x ∈ dom τt . By Lemma 2.1 the map τ˜[s] is well-defined. It is clearly bijective, as each τt , t ∈ [s], is a bijection. Proposition 2.3. Assume the conditions of Lemma 2.1. Then τ˜ is a partial action of S/ρ on X. Proof. We first show that τ˜[s−1 ] = τ˜−1 [s] . Observe that x ∈ dom τ˜[s−1 ] ⇔ x ∈ dom τt for some t ∈ [s−1 ] ⇔ x ∈ ran τu for some u ∈ [s]. Thus, dom τ˜[s−1 ] = ran τ˜[s] = dom τ˜−1 ˜[s−1 ] (x) = τt (x) = τ −1 [s] . Moreover, τ t−1 (x) = −1 −1 −1 τ˜[s] (x), where x ∈ dom τt and t ∈ [s ] (equivalently, x ∈ ran τt−1 and t ∈ [s]). We now prove the inequality τ˜[s] ◦ τ˜[t] ≤ τ˜[st] . Take x ∈ X with x ∈ dom τ˜[t] and τ˜[t] (x) ∈ dom τ˜[s] . There is t′ ∈ [t], such that x ∈ dom τt′ . Then τ˜[t] (x) = τt′ (x) ∈ dom τs′ for some s′ ∈ [s], and hence τ˜[s] ◦ τ˜[t] (x) = τs′ ◦ τt′ (x). Since τs′ ◦ τt′ ≤ τs′ t′ , one has x ∈ dom τs′ t′ and τs′ ◦ τt′ (x) = τs′ t′ (x), the latter being τ[st] (x), as s′ t′ ∈ [st].
PARTIAL ACTIONS AND A THEOREM BY O’CARROLL
3
Remark 2.4. If, under the conditions of Proposition 2.3, S is E-unitary, ρ = σ S and e∈E(S) dom τe = X, then τ˜ is a partial action [3, 4] of the maximum group image G(S) = S/σ of S on X as in [9, Theorem 1.2]. Indeed, we only S need to show that the identity element 1G(S) of G(S) acts trivially: dom τ˜1G(S) = e∈E(S) dom τe = X and τ˜1G(S) (x) = τe (x) = x for x ∈ dom τe . Remark 2.5. If, under the conditions of Proposition 2.3, S is an F -inverse monoid and ρ = σ, then [ [ dom τt = dom τmax[s] dom τt = dom τ˜[s] = t∈[s]
t≤max[s]
and hence τ˜[s] = τmax[s] . In particular, if S is the Exel’s monoid S(G) of a group G, then identifying G(S) with G, we see that τ˜ is the partial action of G on X induced by τ as in [3, Theorem 4.2]. 3. A theorem by O’Carroll Lemma 3.1. Let S be an inverse semigroup, ρ an idempotent pure congruence on S, X a set and τ a partial action of S on X. Then for any (s, t) ∈ ρ τs (dom τs ∩ dom τt ) = τt (dom τs ∩ dom τt ) = ran τs ∩ ran τt . Proof. By Lemma 2.1 τs (dom τs ∩ dom τt ) = τt (dom τs ∩ dom τt ) ⊆ ran τs ∩ ran τt .
(3)
For the converse inclusion replace in (3) the elements s and t by their inverses: −1 τ −1 s (ran τs ∩ ran τt ) = τ t (ran τs ∩ ran τt ) ⊆ dom τs ∩ dom τt .
Let E be a semilattice, i. e. a commutative semigroup, whose elements are idempotents. Denote by Ii (E) (respectively, Ipi (E)) the inverse subsemigroup of I(E) consisting of the isomorphisms between ideals (respectively, principal ideals) of E. By a partial action τ of an inverse semigroup S on E we mean a premorphism τ : S → Ii (E). If τs ∈ Ipi (E) for all s ∈ S, then we say that τ is unital. For any partial action τ of S on E define E ∗τ S to be the set {eδs | s ∈ S and e ∈ −1 ran τs } with the multiplication eδs · f δt = τs (τ −1 s (e)f )δst . Since τs (τ s (e)f ) ∈ τs (dom τs ∩ ran τt ) = ran (τs ◦ τt ) ⊆ ran τst , the product is well-defined. Lemma 3.2. The set E ∗τ S is an inverse semigroup.2 Proof. As in the proof of [1, Theorem 3.1] the associativity of E ∗τ S reduces −1 to the equality τs (τ −1 s (ef )g) = eτs (τ s (f )g), where f ∈ dom τs . The latter is straightforward: −1 −1 −1 −1 τs (τ −1 s (ef )g) = τs (τ s (ef )τ s (f )g) = ef τs (τ s (f )g) = eτs (τ s (f )g).
The idempotents of E ∗τ S are of the form eδf , where f ∈ E(S). Clearly, any two idempotents commute. By a direct calculation one checks that τ s−1 (e)δs−1 is inverse to eδs (see, for example, the proof of [2, Lemma 5.12]). Denote by ρ˜ the congruence {(eδs , eδt ) ∈ (E ∗τ S)2 | (s, t) ∈ ρ} on E ∗τ S. 2The semigroup E ∗ S is an analogue of L(T, X, Y ) from [8] τ
4
MYKOLA KHRYPCHENKO
Proposition 3.3. Let τ be a unital partial action of an inverse semigroup S on a semilattice E and ρ an idempotent pure congruence on S. Then τ˜ is a partial action of S/ρ on the semilattice E and (E ∗τ S)/ρ˜ ∼ = E ∗τ˜ (S/ρ). Proof. We know from Proposition 2.3 that τ˜ S is a premorphism from S/ρ to I(E). Clearly, dom τ˜[s] is an ideal of E as the union t∈[s] dom τt of ideals of E. We need to prove that τ˜[s] is a homomorphism dom τ˜[s] → ran τ˜[s] . By the definition of τ˜ it suffices to show that τs (e)τt (f ) = τs (ef ) = τt (ef )
(4)
for (s, t) ∈ ρ and e ∈ dom τs , f ∈ dom τt . Since τ is unital, for each s ∈ S there is 1s ∈ E, such that ran τs = 1s E. In particular, dom τs = 1s−1 E and dom τt = 1t−1 E. It follows from Lemma 3.1 that τt (1s−1 1t−1 ) = 1s 1t . Therefore, τs (e)τt (f ) = 1s 1t τs (e)τt (f ) = τt (1s−1 1t−1 )τs (e)τt (f ) = τs (e)τt (1s−1 f ). As 1s−1 f ∈ dom τs ∩ dom τt , by Lemma 2.1 one has τt (1s−1 f ) = τs (1s−1 f ) and thus τs (e)τt (1s−1 f ) = τs (1s−1 ef ) = τs (ef ), proving (4). To prove the second statement of the theorem, for any [eδs ] ∈ (E ∗τ S)/ρ˜ define ϕ([eδs ]) = eδ[s] . Observe that the definition does not depend on the representative eδs of the class [eδs ] and gives different eδ[s] for different [eδs ]. Since [ ran τt = ran τ˜[s] , e ∈ ran τs ⊆ t∈[s]
we have eδ[s] ∈ E ∗τ˜ (S/ρ), so ϕ is an injective map from (E ∗τ S)/ρ˜ to E ∗τ˜ (S/ρ). It is also clearly surjective, because e ∈ ran τ˜[s] means that e ∈ ran τt for some t ∈ [s]. It remains to prove that ϕ is a homomorphism: ϕ([eδs ][f δt ]) = ϕ([τs (τ −1 s (e)f )δst ]) = τs (τ −1 s (e)f )δ[st] = τ˜[s] (˜ τ −1 [s] (e)f )δ[s][t] = eδ[s] · f δ[t] = ϕ([eδs ])ϕ([f δt ]). Proposition 3.4. Let S be an inverse semigroup, ρ an idempotent pure congruence on S and τ the canonical (unital) action of S on E(S), namely, τs : s−1 sE(S) → ss−1 E(S) and τs (e) = ses−1 . Then S embeds into (E(S) ∗τ S)/ρ˜. Proof. Define φ : S → (E(S) ∗τ S)/ρ˜ by φ(s) = [ss−1 δs ]. Since ss−1 δs · tt−1 δt = ss−1 · s(tt−1 )s−1 δst = st(st)−1 δst , the map φ is a homomorphism. Suppose that φ(s) = φ(t). By the definition of ρ˜ this means that ss−1 = tt−1 and (s, t) ∈ ρ. The latter yields t−1 s ∈ E(S), as ρ is idempotent pure. Then s = ss−1 s = tt−1 · s = t · t−1 s ≤ t. By symmetry t ≤ s. Hence s = t, and φ is one-to-one.
PARTIAL ACTIONS AND A THEOREM BY O’CARROLL
5
Corollary 3.5. Let S be an inverse semigroup and ρ an idempotent pure congruence on S. Then there exists a semilattice E ∼ = E(S) and a partial action τ of S/ρ on E, such that S embeds into E ∗τ (S/ρ). The embedding is explained by Propositions 3.3 and 3.4. To make the result of Corollary 3.5 more precise, we follow [8]. We say that a partial action τ of S on E is strict, if for each e ∈ E the set {f ∈ E(S) | e ∈ dom τf } has the minimum element, denoted by α(e), and the map α : E → E(S) is a homomorphism. Observe that if eδs ∈ E ∗τ S, then α(e) ≤ ss−1 . The set of eδs ∈ E ∗τ S for which α(e) = ss−1 will be denoted by E ⋆τ S.3 Lemma 3.6. The subset E ⋆τ S is an inverse subsemigroup of E ∗τ S. Proof. We first observe, as in [8, Lemma 2], that α(τs (e)) = sα(e)s−1
(5)
for e ∈ dom τs . Indeed, e ∈ dom τα(e) , therefore, τs (e) ∈ ran (τs ◦ τα(e) ) ⊆ ran τsα(e) ⊆ dom τsα(e)s−1 , so α(τs (e)) ≤ sα(e)s−1 . For the converse inequality replace e by τs (e) and s by s−1 . Let eδs , f δt ∈ E ⋆τ S. Then for their product τs (τ −1 s (e)f )δst we have by (5) −1 −1 α(τs (τ −1 = ss−1 α(e)sα(f )s−1 = stt−1 s−1 . s (e)f )) = sα(τ s (e)f )s
Hence, eδs · f δt ∈ E ⋆τ S. Moreover, for each eδs ∈ E ⋆τ S the inverse (eδs )−1 = −1 −1 τ −1 α(e)s = s−1 s. s (e)δs−1 belongs to E ⋆τ S, as α(τ s (e)) = s A strict partial action τ of S on E is called fully strict, if the homomorphism E ⋆τ S → S, which maps eδs to s, is surjective. Remark 3.7. Let τ be a fully strict partial action of S on E. Then E ⋆τ S = E ∗τ S if and only if S is a group. Indeed, the “if” part trivially holds for any strict τ . For the “only if” part take s ∈ S and suppose that s ≤ t for some t ∈ S. Since τ is fully strict, there are e, f ∈ E, such that eδs , f δt ∈ E ⋆τ S. In particular, e ∈ ran τs and f ∈ ran τt . Since ran τs and ran τt are ideals of E, we have ef ∈ ran τs ∩ ran τt , so ef δs , ef δt ∈ E ∗τ S. But E ∗τ S = E ⋆τ S, so α(ef ) = ss−1 = tt−1 . Thus, s = ss−1 t = tt−1 t = t, and the partial order on S is trivial. Theorem 3.8 (Theorem 4 from [8]). Given an inverse semigroup S and an idempotent pure congruence ρ on S, there exists a semilattice E ∼ = E(S) and a fully strict partial action τ of S/ρ on E, such that S embeds into E ∗τ (S/ρ). Moreover, the image of S in E ∗τ (S/ρ) coincides with E ⋆τ (S/ρ). In particular, the embedding is surjective if and only if ρ is a group congruence, i. e. S is E-unitary and ρ = σ. Proof. As we know from Propositions 3.3 and 3.4, E can be chosen to be equal to E(S) and [ [ tt−1 E(S), t−1 tE(S) → τ[s] : t∈[s]
t∈[s]
τ[s] (e) = tet−1 for e ∈ t−1 tE(S). Consider the set Ae = {[f ] ∈ E(S/ρ) | e ∈ dom τ[f ] }. Clearly, [e] ∈ Ae , as e ∈ eE(S) ⊆ dom τ[e] . Moreover, if [f ] ∈ Ae , 3This is an analogue of L (T, X, Y ) from [8] m
6
MYKOLA KHRYPCHENKO
then e ∈ gE(S) for some idempotent g ∈ [f ]. So, e ≤ g and hence [e] ≤ [g] = [f ]. This shows that [e] is the minimum element of Ae , and since α(e) = [e] is a homomorphism E(S) → E(S/ρ), the partial action τ is strict. For each [s] ∈ S/ρ the element ss−1 δ[s] belongs to E(S) ∗τ (S/ρ), as ss−1 ∈ ss−1 E(S) ⊆ ran τ[s] . Moreover, α(ss−1 ) = [ss−1 ] = [s][s]−1 , so in fact ss−1 δ[s] ∈ E(S) ⋆τ (S/ρ), proving that τ is fully strict. The embedding S → E(S) ∗τ (S/ρ) maps s to ss−1 δ[s] , which is an element of E(S) ⋆τ (S/ρ) as shown above. Now if eδ[s] ∈ E(S) ⋆τ (S/ρ), then e ∈ tt−1 E(S) for some t ∈ [s] and [e] = α(e) = [s][s]−1 = [ss−1 ]. It follows that e = ett−1 = et(et)−1 and [et] = [ss−1 ][t] = [ss−1 ][s] = [s]. Hence, eδ[s] is the image of et ∈ S. Finally, if the embedding is surjective, then E(S) ⋆τ (S/ρ) = E(S) ∗τ (S/ρ), so S/ρ is a group by Remark 3.7. References [1] Dokuchaev, M., and Exel, R. Associativity of crossed products by partial actions, enveloping actions and partial representations. Trans. Amer. Math. Soc. 357, 5 (2005), 1931–1952. [2] Dokuchaev, M., and Khrypchenko, M. Twisted partial actions and extensions of semilattices of groups by groups. arXiv:1602.02424 (2016). [3] Exel, R. Partial actions of groups and actions of inverse semigroups. Proc. Amer. Math. Soc. 126, 12 (1998), 3481–3494. [4] Kellendonk, J., and Lawson, M. V. Partial actions of groups. Internat. J. Algebra Comput. 14, 1 (2004), 87–114. [5] Lawson, M. V., and Margolis, S. W. In McAlister’s footsteps: a random ramble around the P -theorem. In Semigroups and formal languages. World Sci. Publ., Hackensack, NJ, 2007, pp. 145–163. [6] Lawson, M. V., Margolis, S. W., and Steinberg, B. Expansions of inverse semigroups. J. Aust. Math. Soc. 80, 2 (2006), 205–228. [7] McAlister, D. B. Groups, semilattices and inverse semigroups II. Trans. Am. Math. Soc. 196 (1974), 351–370. [8] O’Carroll, L. Strongly E-reflexive inverse semigroups. Proc. Edinburgh Math. Soc. (2) 20 (1977), 339–354. [9] Steinberg, B. Inverse semigroup homomorphisms via partial group actions. Bull. Aust. Math. Soc. 64, 1 (2001), 157–168. ´ tica, Universidade Federal de Santa Catarina, Campus ReDepartamento de Matema ˜ o David Ferreira Lima, Floriano ´ polis, SC, CEP: 88040–900, Brazil itor Joa E-mail address:
[email protected]
