aa r X i v : . [ m a t h . G R ] D ec ON MONOIDS OF INJECTIVE PARTIAL COFINITESELFMAPS
OLEG GUTIK* AND DUˇSAN REPOVˇS**
Abstract.
We study the semigroup I cf λ of injective partial cofinite self-maps of an infinite cardinal λ . We show that I cf λ is a bisimple inversesemigroup and each chain of idempotents in I cf λ is contained in a bicyclicsubsemigroup of I cf λ , we describe the Green relations on I cf λ and we provethat every non-trivial congruence on I cf λ is a group congruence. Also, wedescribe the structure of the quotient semigroup I cf λ /σ , where σ is the leastgroup congruence on I cf λ . Introduction and preliminaries
In this paper we shall denote the first infinite ordinal by ω and the cardinalityof the set A by | A | . We shall identify all cardinals with their correspondinginitial ordinals. We shall denote the set of integers by Z and the additive groupof integers by Z (+).A semigroup S is called inverse if for every element x ∈ S there exists aunique x − ∈ S such that xx − x = x and x − xx − = x − . The element x − is called the inverse of x ∈ S . If S is an inverse semigroup, then the functioninv : S → S which assigns to every element x of S its inverse element x − iscalled the inversion .A congruence C on a semigroup S is called non-trivial if C is distinct fromthe universal and the identity congruences on S , and a group congruence if thequotient semigroup S/ C is a group.If S is a semigroup, then we shall denote the subset of all idempotents in S by E ( S ). If S is an inverse semigroup, then E ( S ) is closed under multiplicationand we shall refer to E ( S ) as a band (or the band of S ). Then the semigroupoperation on S determines the following partial order on E ( S ): e f if andonly if ef = f e = e . This order is called the natural partial order on E ( S ). Mathematics Subject Classification.
Primary 20M20, 20M18; Secondary 20B30.
Key words and phrases.
Bicyclic semigroup, semigroup of bijective partial transformations,congruence, symmetric group, group congruence, semidirect product.This research was supported by the Slovenian Research Agency grants P1-0292-0101, J1-5435-0101 and J1-6721-0101. A semilattice is a commutative semigroup of idempotents. A semilattice E iscalled linearly ordered or a chain if its natural order is a linear order. A maximalchain of a semilattice E is a chain which is not properly contained in any otherchain of E .The Axiom of Choice implies the existence of maximal chains in every par-tially ordered set. According to [13, Definition II.5.12], a chain L is called an ω -chain if L is isomorphic to { , − , − , − , . . . } with the usual order or equiv-alently, if L is isomorphic to ( ω, max). Let E be a semilattice and e ∈ E . Weput ↓ e = { f ∈ E | f e } and ↑ e = { f ∈ E | e f } . By ( P <ω ( λ ) , ∪ ) weshall denote the free semilattice with identity over a set of cardinality λ > ω ,i.e., ( P <ω ( λ ) , ∪ ) is the set of all finite subsets (with the empty set) of λ with thesemilattice operation “union”.If S is a semigroup, then we shall denote the Green relations on S by R , L , J , D and H (see [5]). A semigroup S is called simple if S does not contain propertwo-sided ideals and bisimple if S has only one D -class.The bicyclic semigroup C ( p, q ) is the semigroup with the identity 1 generatedby elements p and q subject only to the condition pq = 1. The bicyclic semi-group is bisimple and every one of its congruences is either trivial or a groupcongruence. Moreover, every homomorphism h of the bicyclic semigroup is ei-ther an isomorphism or the image of C ( p, q ) under h is a cyclic group (see [5,Corollary 1.32]). The bicyclic semigroup plays an important role in algebraictheory of semigroups and in the theory of topological semigroups. For examplea well-known Andersen’s result [1] states that a (0–)simple semigroup is com-pletely (0–)simple if and only if it does not contain the bicyclic semigroup. Thebicyclic semigroup admits only the discrete topology [7]. The problem of em-beddability of the bicycle semigroup into compact-like semigroups was studiedin [2, 3, 4, 8, 11]. Remark 1.
We observe that the bicyclic semigroup is isomorphic to the semi-group C N ( α, β ) which is generated by partial transformations α and β of the setof positive integers N , defined as follows: ( n ) α = n + 1 if n > n ) β = n − n > ii ) in [13]).If T is a semigroup, then we say that a subsemigroup S of T is a bicyclicsubsemigroup of T if S is isomorphic to the bicyclic semigroup C ( p, q ).Hereafter we shall assume that λ is an infinite cardinal. If α : X ⇀ Y is apartial map, then we shall denote the domain and the range of α by dom α andran α , respectively.Let I λ denote the set of all partial one-to-one transformations of an infiniteset X of cardinality λ together with the following semigroup operation: x ( αβ ) = ( xα ) β if x ∈ dom( αβ ) = { y ∈ dom α | yα ∈ dom β } , for α, β ∈ I λ . N MONOIDS OF INJECTIVE PARTIAL COFINITE SELFMAPS 3
The semigroup I λ is called the symmetric inverse semigroup over the set X (see[5, Section 1.9]). The symmetric inverse semigroup was introduced by Vagner [21]and it plays a major role in the theory of semigroups.Furthermore, we shall identify the cardinal λ = | X | with the set X . By I cf λ we shall denote a subsemigroup of injective partial selfmaps of λ with cofinitedomains and ranges in I λ , i.e., I cf λ = { α ∈ I λ | | λ \ dom α | < ∞ and | λ \ ran α | < ∞} . Obviously, I cf λ is an inverse submonoid of the semigroup I λ . We shall call thesemigroup I cf λ the monoid of injective partial cofinite selfmaps of λ .Next, by I we shall denote the identity and by H ( I ) the group of units of thesemigroup I cf λ .It well known that each partial injective cofinite selfmap f of λ induces ahomeomorphism f ∗ : λ ∗ → λ ∗ of the remainder λ ∗ = βλ \ λ of the Stone- ˇCechcompactification of the discrete space λ . Moreover, under some set theoreticaxioms (like PFA or OCA ), each homeomorphism of ω ∗ is induced by somepartial injective cofinite selfmap of ω (see [15]–[20]). So the inverse semigroup I cf λ admits a natural homomorphism h : I cf λ → H ( λ ∗ ) to the homeomorphism group H ( λ ∗ ) of λ ∗ and this homomorphism is surjective under certain set theoreticassumptions.The semigroups I ր∞ ( N ) and I ր∞ ( Z ) of injective isotone partial selfmaps withcofinite domains and images of positive integers and integers, respectively, werestudied in [9] and [10]. There it was proved that the semigroups I ր∞ ( N ) and I ր∞ ( Z ) have properties similar to the bicyclic semigroup: they are bisimple andevery non-trivial homomorphic image of I ր∞ ( N ) and I ր∞ ( Z ) is a group, and more-over, the semigroup I ր∞ ( N ) has Z (+) as a maximal group image and I ր∞ ( Z ) has Z (+) × Z (+), respectively.In this paper we shall study algebraic properties of the semigroup I cf λ . We shallshow that I cf λ is a bisimple inverse semigroup and every chain of idempotents in I cf λ is contained in a bicyclic subsemigroup of I cf λ , we shall describe the Greenrelations on I cf λ and we shall prove that every non-trivial congruence on I cf λ is a group congruence. Also, we shall describe the structure of the quotientsemigroup I cf λ /σ , where σ is the least group congruence on I cf λ .2. Algebraic properties of the semigroup I cf λ Proposition 2.1. ( i ) I cf λ is a simple semigroup. ( ii ) An element α of the semigroup I cf λ is an idempotent if and only if ( x ) α = x for every x ∈ dom α . ( iii ) If ε, ι ∈ E ( I cf λ ) , then ε ι if and only if dom ε ⊆ dom ι . ( iv ) The semilattice E ( I cf λ ) is isomorphic to ( P <ω ( λ ) , ∪ ) under the mapping ( ε ) h = λ \ dom ε . OLEG GUTIK AND DUˇSAN REPOVˇS ( v ) Every maximal chain in E ( I cf λ ) is an ω -chain. ( vi ) α R β in I cf λ if and only if dom α = dom β . ( vii ) α L β in I cf λ if and only if ran α = ran β . ( viii ) α H β in I cf λ if and only if dom α = dom β and ran α = ran β . ( ix ) α D β for all α, β ∈ I cf λ and hence the semigroup I cf λ is bisimple.Proof. ( i ) We shall show that I cf λ · α · I cf λ = I cf λ for every element α ∈ I cf λ . Let α and β are arbitrary elements of the semigroup I cf λ . We shall choose elements γ, δ ∈ I cf λ such that γ · α · δ = β . We put dom γ = dom β , ran γ = dom α ,dom δ = ran α and ran δ = ran β . Since the sets λ \ dom α and λ \ dom β arefinite we conclude that there exists a bijective map f : dom α → dom β . We put γ = f and ((( x ) γ ) α ) δ = ( x ) β for all x ∈ dom β . Then we have that γ · α · δ = β .Statements ( ii ) − ( v ) are trivial and they follow from the definition of thesemigroup I cf λ . The proofs of ( vi ) − ( viii ) follow trivially from the fact that I cf λ is a regular semigroup, and Proposition 2.4.2 and Exercise 5.11.2 in [12].( ix ) Let α and β be arbitrary elements of the semigroup I cf λ . Since the sets λ \ dom α and λ \ ran β are finite we conclude that there exists a bijective map γ : dom α → ran β . Then γ ∈ I cf λ and by statements ( vi ) and ( vii ) we have that α R γ and β L γ in I cf λ and hence α D β in I cf λ . (cid:3) We denote the group of all bijective transformations of a set of cardinality λ by S λ . Then we get the following: Corollary 2.2.
The group of units H ( I ) of the semigroup I cf λ is isomorphic to S λ . For any idempotents ε and ι of the semigroup I cf λ we denote: H ( ε, ι ) = (cid:8) χ ∈ I cf λ | χ · χ − = ε and χ − · χ = ι (cid:9) and H ( ε ) = H ( ε, ε ) . Proposition 2.1( viii ) implies that the set H ( ε, ι ) is a H -class and the set H ( ε )is a a maximal subgroup in I cf λ for all idempotents ε, ι ∈ I cf λ .Corollary 2.2 and Proposition 2.20 of [5] imply the following: Corollary 2.3.
Every maximal subgroup of the semigroup I cf λ is isomorphicto S λ . Proposition 2.4. (cid:12)(cid:12) I cf λ (cid:12)(cid:12) = 2 | λ | .Proof. Since | λ × λ | = | λ | we have that | S λ | | λ × λ | = 2 | λ | . Since | λ ⊔ λ | = | λ | there exists an injective map f : P ( λ ) → S λ ⊔ λ from the set P ( λ ) of all subsetof the cardinal λ into the group S λ ⊔ λ defined in the following way: f ( A ) is abijection on λ ⊔ λ with support A ⊔ A . Then we have that | S λ | > | λ ⊔ λ | = 2 | λ | and hence | S λ | = 2 | λ | .Since | P <ω ( λ ) | = | P <ω ( λ ) × P <ω ( λ ) | = λ we conclude that Theorem 2.20from [5] and Proposition 2.1( viii ) imply that (cid:12)(cid:12) I cf λ (cid:12)(cid:12) = | P <ω ( λ ) × P <ω ( λ ) × S λ | = | P <ω ( λ ) × P <ω ( λ ) | · | S λ | = | λ | · | λ | = 2 | λ | . N MONOIDS OF INJECTIVE PARTIAL COFINITE SELFMAPS 5 (cid:3)
Proposition 2.5.
For every α, β ∈ I cf λ , both sets (cid:8) χ ∈ I cf λ | α · χ = β (cid:9) and (cid:8) χ ∈ I cf λ | χ · α = β (cid:9) are finite. Consequently, every right translation and every left translation by anelement of the semigroup I cf λ is a finite-to-one map.Proof. We denote A = (cid:8) χ ∈ I cf λ | α · χ = β (cid:9) and B = (cid:8) χ ∈ I cf λ | α − · α · χ = α − · β (cid:9) . Then A ⊆ B and the restriction of any partial map χ ∈ B onto dom( α − · α )coincides with the partial map α − · β . Since every partial map from I cf λ hascofinite range and cofinite domain we conclude that the set B is finite and henceso is A . (cid:3) Proposition 2.6.
Each maximal chain L of idempotents in I cf λ coincides withthe idempotent band E ( S ) of a bicyclic subsemigroup S of I cf λ .Proof. By Proposition 2.1( iii ), the chain L can be written as L = { ε n } ∞ n =1 where ε > ε > · · · > ε n > · · · . Since every infinite subchain of an ω -chain isalso an ω -chain we have that Proposition 2.1( v ) implies that L is an ω -chain.Then by Proposition 2.1( iii ) we get that dom ε i \ dom ε i +1 = ∅ for all positiveintegers i . Also, the maximality of L implies that the set dom ε i \ dom ε i +1 is a singleton for all positive integers i . For every positive integer i we put { x i } = dom ε i \ dom ε i +1 . Then we put D = dom ε \ S i ∈ N { x i } and define thepartial maps α : λ ⇀ λ and β : λ ⇀ λ as follows:( x ) α = (cid:26) x n +1 , if x = x n ∈ dom ε \ D and n > x, if x ∈ D ;and ( x ) β = (cid:26) x n − , if x = x n ∈ dom ε \ D and n > x, if x ∈ D. Since the set λ \ dom ε is finite we have that α, β ∈ I cf λ and Remark 1 impliesthe statement of our proposition. (cid:3) Proposition 2.6 and the Axiom of Choice imply the following proposition.
Proposition 2.7.
Each chain of idempotents in I cf λ is contained in a bicyclicsubsemigroup of I cf λ . Proposition 2.8.
Let C be a congruence on the semigroup I cf λ . If there existtwo non- H -equivalent elements α, β ∈ I cf λ such that α C β , then C is a groupcongruence on I cf λ . OLEG GUTIK AND DUˇSAN REPOVˇS
Proof.
First we suppose that α and β are distinct idempotents of the semigroup I cf λ . Without loss of generality we can assume that α and β are compatible and α β . Otherwise, replace α by α · β . Then by Proposition 2.7 there existsa maximal chain L in E ( I cf λ ) such that L contains the elements α and β , andhence L contained in a bicyclic subsemigroup S of I cf λ . Then Corollary 1.32 of [5]implies that ε C ι for all elements ε and ι of the chain L .Let ν be an arbitrary idempotent of the semigroup I cf λ . Obviously, if ε, ι ∈ L such that ε ι then ε · ν ι · ν . Since ↑ e is a finite subset of the free semilatticewith unity ( P <ω ( λ ) , ⊆ ) for any e ∈ ( P <ω ( λ ) , ⊆ ), we have that Proposition 2.1( iv )implies that νL is an infinite chain in E ( I cf λ ). Then we have that ε C ι for all ε, ι ∈ νL . We put L ν = νL ∪ { ν } ∪ { I } . Then L ν is a chain in E ( I cf λ ). Thereforeby Proposition 2.7 we get that there exists a maximal chain L max in E ( I cf λ ) whichcontains the chain L ν and L max is a band of a bicyclic subsemigroup S in I cf λ .Now Corollary 1.32 of [5] implies that ε C ι for all elements ε and ι of the chain L ν .Hence ν C I and α C I imply that ν C α . Therefore all idempotents of the semigroup I cf λ are C -equivalent. Since the semigroup I cf λ is inverse we conclude that quotientsemigroup I cf λ / C contains only one idempotent and by Lemma II.1.10 from [13]the semigroup I cf λ / C is a group.Suppose that α and β are distinct non– H -equivalent elements of the semi-group I cf λ such that α C β . Then Proposition 2.1 implies that at least one of thefollowing conditions holds: αα − = ββ − or α − α = β − β. By Lemma III.1.1 from [13] we have that α − C β − . Then αα − C αβ − and ββ − C αβ − and hence αα − C ββ − . Similarly we get that α − α C β − β . Thenthe first part of the proof implies that C is a group congruence on I cf λ . (cid:3) Theorem 2.9.
Every non-trivial congruence on the semigroup I cf λ is a groupcongruence.Proof. Let C be a non-trivial congruence on the semigroup I cf λ . Let α and β bedistinct C -equivalent elements of the semigroup I cf λ . If the elements α and β arenot H -equivalent then Proposition 2.8 implies the statement of the theorem.Suppose that α H β . Then Theorem 2.20 from [5] implies that without loss ofgenerality we can assume that α and β are elements of the group of units H ( I )of the semigroup I cf λ and hence I C ( βα − ). We denote γ = βα − . Since I = γ we conclude that there exists x ∈ λ such that ( x ) γ = x . We define ε to bean identity selfmap of the set λ \ { x } . Then ε ∈ I cf λ and ( ε · I ) C ( ε · γ ). Since( x ) γ = x we have that Proposition 2.1( viii ) implies that the elements ε and ε · γ are not H -equivalent. Then by Proposition 2.8 we get that C is a groupcongruence on I cf λ . (cid:3) N MONOIDS OF INJECTIVE PARTIAL COFINITE SELFMAPS 7 On the least group congruence on the semigroup I cf λ Every inverse semigroup S admits the least group congruence σ (see [13,Section III]): sσt if and only if there exists an idempotent e ∈ S such that se = te. Theorem 2.9 implies that every non-injective homomorphism h : I cf λ → S fromthe semigroup I cf λ into an arbitrary semigroup S generates a group congruence h on I cf λ . In this section we describe the structure of the quotient semigroup I cf λ /σ . Proposition 3.1. If ασβ in I cf λ then | λ \ dom α | − | λ \ ran α | = | λ \ dom β | − | λ \ ran β | . Proof.
Let ε be an idempotent of the semigroup I cf λ such that αε = βε . We shallshow that the statement of the proposition holds for the elements α and αε .Without loss of generality we can assume that ε α − α , i.e., dom ε ⊆ dom( α − α ). Since α is an injective partial map with | λ \ dom α | < ∞ and | λ \ ran α | < ∞ , and ε is an identity map of the cofinite subset dom ε in λ weconclude that | λ \ dom α | − | λ \ ran α | = | λ \ dom( αε ) | − | λ \ ran( αε ) | . This implies the statement of the proposition. (cid:3)
For an arbitrary element α of the semigroup I cf λ we denote d ( α ) = | λ \ dom α | and r ( α ) = | λ \ ran α | . Proposition 3.2. If α and β are arbitrary elements of the semigroup I cf λ then d ( αβ ) − r ( αβ ) = d ( α ) − r ( α ) + d ( β ) − r ( β ) . Proof.
We consider four cases.(1) First we consider the case when ran α ⊆ dom β . We put k = r ( α ) − d ( β ).Then the definition of the semigroup I cf λ implies that k > d ( αβ ) = d ( α ), r ( αβ ) = r ( β ) − k , and hence in this case we get that d ( αβ ) − r ( αβ ) = d ( α ) − r ( α ) + d ( β ) − r ( β ) . (2) Suppose that the case when dom β ⊆ ran α holds. We put k = d ( β ) − r ( α ).Then the definition of the semigroup I cf λ implies that k > d ( αβ ) = d ( α ) + k , r ( αβ ) = r ( β ), and hence in this case we have that d ( αβ ) − r ( αβ ) = d ( α ) − r ( α ) + d ( β ) − r ( β ) . (3) Now we consider the case ( λ \ ran α ) ∩ ( λ \ dom β ) = ∅ , ran α * dom β and dom β * ran α . Then the definition of the semigroup I cf λ implies that thereexist positive integers i , j and k such that i = | ( λ \ ran α ) \ ( λ \ dom β ) | , j = | ( λ \ ran α ) ∩ ( λ \ dom β ) | and k = | ( λ \ dom β ) \ ( λ \ ran α ) | . Then we have OLEG GUTIK AND DUˇSAN REPOVˇS that r ( α ) = i + j , d ( β ) = j + k , d ( αβ ) = d ( α ) + k and r ( αβ ) = r ( β ) + i .Therefore, in this case we get that d ( αβ ) − r ( αβ ) = d ( α ) − r ( α ) + d ( β ) − r ( β ) . (4) In the case when ( λ \ ran α ) ∩ ( λ \ dom β ) = ∅ we have that the definitionof the semigroup I cf λ implies that d ( αβ ) = d ( α ) + d ( β ), r ( αβ ) = r ( α ) + r ( β ), andhence we get that d ( αβ ) − r ( αβ ) = d ( α ) − r ( α ) + d ( β ) − r ( β ) . This completes the proof of the proposition. (cid:3)
On the semigroup I cf λ we define a relation ∼ d in the following way: α ∼ d β if and only if d ( α ) − r ( α ) = d ( β ) − r ( β ) , for α, β ∈ I cf λ . Proposition 3.3.
Let λ be an infinite cardinal. Then ∼ d is a congruence onthe semigroup I cf λ and moreover the quotient semigroup I cf λ / ∼ d is isomorphic tothe additive group of integers Z (+) .Proof. Simple verifications and Proposition 3.2 imply that ∼ d is a congruenceon the semigroup I cf λ . We define a homomorphism h : I cf λ → Z (+) by the formula( α ) h = d ( α ) − r ( α ). Then the definitions of the semigroup I cf λ and the congruence ∼ d on I cf λ , and Proposition 3.2 imply that thus defined map h is a surjectivehomomorphism and moreover ( α ) h = ( β ) h if and only if α ∼ d β in I cf λ . Thiscompletes the proof of the proposition. (cid:3) Proposition 3.4.
Let λ be an infinite cardinal. Then for every element β ofthe semigroup I cf λ such that d ( β ) = r ( β ) there exists an element α of the groupof units of I cf λ such that ασβ .Proof. Fix an arbitrary element β of the semigroup I cf λ . Without loss of gener-ality we can assume that d ( β ) = r ( β ) = k >
0. Let { x , . . . , x k } = λ \ dom β and { y , . . . , y k } = λ \ ran β . We define a map α : λ → λ in the following way:( x ) α = (cid:26) ( x ) β, if x ∈ dom β ; y i , if x = x i , i = 1 , . . . , k. Then α is an element of the group of units of the semigroup I cf λ and it is obviouslythat αε = βε , where ε is the identity map of the set ran β . (cid:3) For every α ∈ S λ we denote supp( α ) = { x ∈ λ | ( x ) α = x } . We define S ∞ λ = { α ∈ S λ | supp( α ) is finite } . We observe that the Schreier–Ulam theorem (see [14, Theorem 11.3.4]) impliesthat S ∞ λ is a normal subgroup of S λ and hence S λ / S ∞ λ is a group. N MONOIDS OF INJECTIVE PARTIAL COFINITE SELFMAPS 9
Later on, when C is a congruence on a semigroup S we shall denote the naturalhomomorphism generated by the congruence C on S by π C : S → S/ C .The definition of the least group congruence σ on the semigroup I cf λ impliesthe following proposition. Proposition 3.5.
Let λ be an infinite cardinal. Then the homomorphic image ( H ( I )) π σ of the group of units H ( I ) of I cf λ under the natural homomorphism π σ : I cf λ → I cf λ /σ is isomorphic to the quotient group S λ / S ∞ λ . Theorem 3.6.
Let λ be an infinite cardinal. Then the following conditions hold: ( i ) ( H ( I )) π σ = S λ / S ∞ λ is a normal subgroup of the group I cf λ /σ ; ( ii ) The group I cf λ /σ contains the infinite cyclic subgroup G (i.e., the additivegroup of integers Z (+) ) such that G ∩ S λ / S ∞ λ = { e } , where e is the unitof the group I cf λ /σ ; ( iii ) S λ / S ∞ λ · G = I cf λ /σ .and hence the group I cf λ /σ is isomorphic to the semidirect product S λ / S ∞ λ ⋉Z (+) .Proof. ( i ) Since σ ⊆∼ d we conclude that Theorem 1.6 of [5] implies that thereexists a unique homomorphism g : I cf λ /σ → G such that the following diagram I cf λ π σ / / π ∼ d " " ❉❉❉❉❉❉❉❉❉ I cf λ /σ g (cid:15) (cid:15) G commutes. Then by Proposition 3.5 we have that the homomorphic image( H ( I )) π σ of the group of units H ( I ) of I cf λ under the natural homomorphism π σ : I cf λ → I cf λ /σ is isomorphic the the quotient group S λ / S ∞ λ . Now Proposi-tions 3.4 and 3.5 imply that the subgroup ( H ( I )) π σ = S λ / S ∞ λ of the group I cf λ /σ is the kernel of the homomorphism g : I cf λ /σ → G , and hence ( H ( I )) π σ = S λ / S ∞ λ is a normal subgroup of I cf λ /σ .( ii ) Fix an arbitrary α ∈ I cf λ such that | λ \ dom α | = 1 and ran α = λ . Thenthe definition of the congruence ∼ d on I cf λ implies that the element α n is not ∼ d -equivalent to any element of the group of units H ( I ) for every non-zero integer n , and hence by Proposition 3.5 we get that (( α ) π σ ) n / ∈ S λ / S ∞ λ . This impliesthat { (( α ) π ∼ d ) n | n ∈ Z } ∩ S λ / S ∞ λ = { e } , where e is the unit of the group I cf λ /σ .Also, it is obvious that ( α n ) π ∼ d = n ∈ G and { ( α n ) π ∼ d | n ∈ Z } is a cyclicsubgroup of S λ / S ∞ λ .( iii ) Fix an arbitrary element x in I cf λ /σ . Let ξ be an arbitrary element of thesemigroup I cf λ be such that ( ξ ) π σ = x . If d ( ξ ) = r ( ξ ) then by Proposition 3.4we have that ξσβ for some element β from the group of units of I cf λ , and hencewe get that x = ( β ) π σ · e ∈ S λ / S ∞ λ · G , where e is the unit of the group I cf λ /σ .Suppose that d ( ξ ) − r ( ξ ) = n = 0. Then by Proposition 3.2 we have that d (cid:0) ξ · ( α − ) n (cid:1) − r (cid:0) ξ · ( α − ) n (cid:1) = 0. Now, Proposition 3.4 implies that the element ξ · ( α − ) n is σ -equivalent to some element β of the group of units H ( I ) of I cf λ .Then we have that ( ξ · ( α − ) n ) π σ = ( β ) π σ and since I cf λ /σ is a group we get that x = ( ξ ) π σ = ( β ) π σ · ( α n ) π σ ∈ S λ / S ∞ λ · G . This implies that S λ / S ∞ λ · G = I cf λ /σ .The last statement of the theorem follows from statements ( i )–( iii ) and Ex-ercise 2.5.3 from [6]. (cid:3) Remark 2.
Proposition 3.3 implies that for every infinite cardinal λ the group I cf λ /σ has infinitely many normal subgroups and hence the semigroup I cf λ hasinfinitely many group congruences. Acknowledgement.
The authors are grateful to the referee for several usefulcomments and suggestions.
References [1] ANDERSEN, O.:
Ein Bericht ¨uber die Struktur abstrakter Halbgruppen , PhD Thesis,Hamburg, 1952.[2] ANDERSON, L. W.—HUNTER, R. P.—KOCH, R. J.:
Some results on stability in semi-groups , Trans. Amer. Math. Soc. (1965), 521–529.[3] BANAKH, T.—DIMITROVA, S.—GUTIK, O.:
The Rees-Suschkiewitsch theorem forsimple topological semigroups , Mat. Stud. (2009), 211–218.[4] BANAKH, T.—DIMITROVA, S.—GUTIK, O.: Embedding the bicyclic semigroup intocountably compact topological semigroups , Topology Appl. (2010), 2803–2814.[5] CLIFFORD, A. H.—PRESTON, G. B.:
The Algebraic Theory of Semigroups , Vol. I/II,American Mathematical Society, Providence, Rhode Island, 1961/1967.[6] DIXON J. D.—MORTIMER B.:
Permutation Groups , Springer, Berlin, 1996.[7] EBERHART, C.—SELDEN, J.:
On the closure of the bicyclic semigroup , Trans. Amer.Math. Soc. (1969), 115–126.[8] GUTIK, O.—REPOVˇS, D.:
On countably compact -simple topological inverse semi-groups , Semigroup Forum (2007), 464–469.[9] GUTIK, O.—REPOVˇS, D.: Topological monoids of monotone injective partial selfmapsof N with cofinite domain and image , Stud. Sci. Math. Hung. (2011), 342–353.[10] GUTIK, O.—REPOVˇS, D.: On monoids of injective partial selfmaps of integers withcofinite domains and images , Georgian Math. J. (2012), 511–532.[11] HILDEBRANT, J. A.—KOCH, R. J.: Swelling actions of Γ -compact semigroups , Semi-group Forum (1986), 65–85.[12] HOWIE, J. M.: Fundamentals of Semigroup Theory , London Math. Soc. Monogr. (N.S.)No. 12, Clarendon Press, Oxford, 1995.[13] PETRICH, M.:
Inverse Semigroups , Pure Appl. Math., John Wiley & Sons, New York,1984.[14] SCOTT, W. R.:
Group Theory , Dover Publications, Inc., New York, 1987.[15] SHELAH, S.—STEPR ¯ANS, J.:
Non-trivial homeomorphisms of βN \ N without the Con-tinuum Hypothesis , Fund. Math. (1989), 135–141.[16] SHELAH, S.—STEPR ¯ANS, J.: Somewhere trivial autohomeomorphisms , J. LondonMath. Soc. (2), (1994), 569–580.[17] SHELAH, S.—STEPR ¯ANS, J.: Martin’s axiom is consistent with the existence ofnowhere trivial automorphisms , Proc. Amer. Math. Soc. (2002), 2097–2106.
N MONOIDS OF INJECTIVE PARTIAL COFINITE SELFMAPS 11 [18] VELIˇCKOVI´C, B.:
Definable automorphisms of P ( ω ) / fin, Proc. Amer. Math. Soc. (1986), 130–135.[19] VELIˇCKOVI´C, B.: Applications of the Open Coloring Axiom , In Set Theory of theContinuum, H. Judah, W. Just et H. Woodin, eds., Pap. Math. Sci. Res. Inst. Workshop,Berkeley, 1989, MSRI Publications. Springer-Verlag. Vol. , Berlin, (1992), pp. 137–154.[20] VELIˇCKOVI´C, B.: OCA and automorphisms of P ( ω ) / fin, Topology Appl. (1993),1–13.[21] VAGNER, V. V.: Generalized groups , Dokl. Akad. Nauk SSSR (1952), 1119–1122 (inRussian). * Department of Mechanics and MathematicsIvan Franko National University of LvivUniversytetska 1Lviv, 79000UKRAINE E-mail address : o [email protected], [email protected] ** Faculty of Education, andFaculty of Mathematics and PhysicsUniversity of LjubljanaKardeljeva ploˇsˇcad 16Ljubljana, 1000SLOVENIA E-mail address ::