On monoids of injective partial selfmaps almost everywhere the identity
aa r X i v : . [ m a t h . G R ] S e p ON MONOIDS OF INJECTIVE PARTIAL SELFMAPS ALMOST EVERYWHERETHE IDENTITY
IVAN CHUCHMAN AND OLEG GUTIK
Abstract.
In this paper we study the semigroup I ∞ λ of injective partial selfmaps almost everywherethe identity of a set of infinite cardinality λ . We describe the Green relations on I ∞ λ , all (two-sided)ideals and all congruences of the semigroup I ∞ λ . We prove that every Hausdorff hereditary Bairetopology τ on I ∞ ω such that ( I ∞ ω , τ ) is a semitopological semigroup is discrete and describe the closureof the discrete semigroup I ∞ λ in a topological semigroup. Also we show that for an infinite cardinal λ the discrete semigroup I ∞ λ does not embed into a compact topological semigroup and construct twonon-discrete Hausdorff topologies turning I ∞ λ into a topological inverse semigroup. Introduction and preliminaries
In this paper all spaces are assumed to be Hausdorff. Furthermore we shall follow the terminologyof [3, 5, 7, 9, 23]. By ω we shall denote the first infinite cardinal and by | A | the cardinality of theset A . If Y is a subspace of a topological space X and A ⊆ Y , then by cl Y ( A ) and Int Y ( A ) we shalldenote the topological closure and the interior of A in Y , respectively.For a semigroup S we denote the semigroup S with the adjoined unit by S (see [5]).An algebraic semigroup S is called inverse if for any element x ∈ S there exists a unique element x − ∈ S (called the inverse of x ) such that xx − x = x and x − xx − = x − . If S is an inverse semigroup,then the function inv : S → S which assigns to every element x of S its inverse element x − is called inversion .If S is an inverse semigroup, then by E ( S ) we shall denote the band (i.e., the subsemigroup ofidempotents) of S . If the band E ( S ) is a non-empty subset of S , then the semigroup operation on S determines a partial order on E ( S ): e f if and only if ef = f e = e . This order is called natural . A semilattice is a commutative semigroup of idempotents. A semilattice E is called linearly ordered or a chain if the semilattice operation induces a linear natural order on E . A maximal chain of a semilattice E is a chain which is properly contained in no other chain of E . The Axiom of Choice implies theexistence of maximal chains in any partially ordered set. According to [21, Definition II.5.12] a chain L is called an ω -chain if L is isomorphic to { , − , − , − , . . . } with the usual order . Let E be asemilattice and e ∈ E . We denote ↓ e = { f ∈ E | f e } and ↑ e = { f ∈ E | e f } . By ( P <ω ( λ ) , ∪ )we shall denote the free semilattice with identity over a cardinal λ > ω , i.e., P <ω ( λ ) is the set of allfinite subsets of λ with the binary operation a · b = a ∪ b , for a, b ∈ P <ω ( λ ).If S is a semigroup, then we shall denote by R , L , J , D and H the Green relations on S (see [5]): a R b if and only if aS = bS ; a L b if and only if S a = S b ; a J b if and only if S aS = S bS ; D = L ◦ R = R ◦ L ; H = L ∩ R . Date : November 20, 2018.2010
Mathematics Subject Classification.
Primary 22A15, 20M20. Secondary 06F30, 20M18, 22A26, 54E52, 54H12,54H15.
Key words and phrases.
Topological semigroup, semitopological semigroup, topological inverse semigroup, semigroupof bijective partial transformations, symmetric inverse semigroup, free semilattice, ideal, congruence, semigroup withthe F -property, Baire space, hereditary Baire space, embedding. The relation J induced a quasi-order J on S as follows: a J b if and only if S aS ⊆ S bS , for a, b ∈ S . This implies that the inclusion order among two-sided ideals of S induces a partial orderamong the J -equivalence classes: J a J b if and only if S aS ⊆ S bS , for a, b ∈ S , where by J a we denote the J -class in S which contains an element a ∈ S (see [17,Section 2.1]). Then we may thus regard S/ J with the relation as a partially ordered set.A semigroup S is called simple if S does not contain proper two-sided ideals.A semitopological (resp. topological ) semigroup is a topological space together with a separately(resp. jointly) continuous semigroup operation. An inverse topological semigroup with the continuousinversion is called a topological inverse semigroup .In the remainder of the paper λ denotes an infinite cardinal.Let I λ denote the set of all partial one-to-one transformations of an infinite cardinal λ togetherwith the following semigroup operation: x ( αβ ) = ( xα ) β if x ∈ dom( αβ ) = { y ∈ dom α | yα ∈ dom β } ,for α, β ∈ I λ . The semigroup I λ is called the symmetric inverse semigroup over the cardinal λ (see[5]). The symmetric inverse semigroup was introduced by Wagner [25] and it plays a major role in thetheory of semigroups.A partial map α ∈ I λ is called almost everywhere the identity if the set λ \ dom α is finite and( x ) α = x only for finitely many x ∈ λ . We denote I ∞ λ = { α ∈ I λ | α is almost everywhere the identity } . Obviously, I ∞ λ is an inverse subsemigroup of the semigroup I ω . The semigroup I ∞ λ is called thesemigroup of injective partial selfmaps almost everywhere the identity of λ . We shall denote everyelement α of the semigroup I ∞ λ by (cid:18) x · · · x n y · · · y n (cid:12)(cid:12)(cid:12)(cid:12) A (cid:19) and this means that the following conditions hold:( i ) A is the maximal subset of λ with the finite complement such that α | A : A → A is an identitymap;( ii ) { x , . . . , x n } and { y , . . . , y n } are finite (not necessary non-empty) subsets of λ \ A ; and( iii ) α maps x i into y i for all i = 1 , . . . , n .We denote the identity of the semigroup I ∞ λ by I .Many semigroup theorists have considered topological semigroups of (continuous) transformationsof topological spaces. Be˘ıda [2], Orlov [19, 20], and Subbiah [24] have considered semigroup and inversesemigroup topologies on semigroups of partial homeomorphisms of some classes of topological spaces.Gutik and Pavlyk [12] considered the special case of the semigroup I nλ : an infinite topologicalsemigroup of λ × λ -matrix units B λ . They showed that an infinite topological semigroup of λ × λ -matrix units B λ does not embed into a compact topological semigroup and that B λ is algebraically h -closed in the class of topological inverse semigroups. They also described the Bohr compactificationof B λ , minimal semigroup and minimal semigroup inverse topologies on B λ .Gutik, Lawson and Repovˇs [11] introduced the notion of a semigroup with a tight ideal seriesand investigated their closures in semitopological semigroups, in particular, in inverse semigroupswith continuous inversion. As a corollary they showed that the symmetric inverse semigroup of finitetransformations I nλ of infinite cardinal λ is algebraically closed in the class of (semi)topological inversesemigroups with continuous inversion. They also derived related results about the nonexistence of(partial) compactifications of semigroups with a tight ideal series.Gutik and Reiter [14] showed that the topological inverse semigroup I nλ is algebraically h -closedin the class of topological inverse semigroups. They also proved that a topological semigroup S with N MONOIDS OF INJECTIVE PARTIAL SELFMAPS ALMOST EVERYWHERE THE IDENTITY 3 countably compact square S × S does not contain the semigroup I nλ for infinite cardinals λ and showedthat the Bohr compactification of an infinite topological semigroup I nλ is the trivial semigroup.In [15] Gutik and Reiter showed that that the symmetric inverse semigroup of finite transformations I nλ of infinite cardinal λ is algebraically closed in the class of semitopological inverse semigroups withcontinuous inversion. Also there they described all congruences on the semigroup I nλ and all compactand countably compact topologies τ on I nλ such that ( I nλ , τ ) is a semitopological semigroup.Gutik, Pavlyk and Reiter [13] showed that a topological semigroup of finite partial bijections I nλ ofan infinite cardinal with a compact subsemigroup of idempotents is absolutely H -closed. They provedthat no Hausdorff countably compact topological semigroup and no Tychonoff topological semigroupwith pseudocompact square contain I nλ as a subsemigroup. They proved that every continuous homo-morphism from a topological semigroup I nλ into a Hausdorff countably compact topological semigroupor Tychonoff topological semigroup with pseudocompact square is annihilating. They also gave suffi-cient conditions for a topological semigroup I λ to be non- H -closed and showed that the topologicalinverse semigroup I λ is absolutely H -closed if and only if the band E ( I λ ) is compact [13].In [16] Gutik and Repovˇs studied the semigroup I ր∞ ( N ) of partial cofinite monotone bijective trans-formations of the set of positive integers N . They showed that the semigroup I ր∞ ( N ) has algebraicproperties similar to the bicyclic semigroup: it is bisimple and all of its non-trivial group homomor-phisms are either isomorphisms or group homomorphisms. They proved that every locally compacttopology τ on I ր∞ ( N ) such that ( I ր∞ ( N ) , τ ) is a topological inverse semigroup, is discrete and describedthe closure of ( I ր∞ ( N ) , τ ) in a topological semigroup.In [4] Gutik and Chuchman studied the semigroup I (cid:31) ր∞ ( N ) of partial co-finite almost monotonebijective transformations of the set of positive integers N . They showed that the semigroup I (cid:31) ր∞ ( N )has algebraic properties similar to the bicyclic semigroup: it is bisimple and all of its non-trivial grouphomomorphisms are either isomorphisms or group homomorphisms. Also they proved that every Bairetopology τ on I (cid:31) ր∞ ( N ) such that ( I (cid:31) ր∞ ( N ) , τ ) is a semitopological semigroup is discrete, described theclosure of ( I (cid:31) ր∞ ( N ) , τ ) in a topological semigroup and constructed non-discrete Hausdorff semigrouptopologies on the semigroup I (cid:31) ր∞ ( N ).In this paper we study the semigroup I ∞ λ of injective partial selfmaps almost everywhere the identityof a set of infinite cardinality λ . We describe the Green relations on I ∞ λ , all (two-sided) ideals and allcongruences of the semigroup I ∞ λ . We prove that every Hausdorff hereditary Baire topology τ on I ∞ ω such that ( I ∞ ω , τ ) is a semitopological semigroup is discrete and describe the closure of the discretesemigroup I ∞ λ in a topological semigroup. Also we show that for an infinite cardinal λ the discretesemigroup I ∞ λ does not embed into a compact topological semigroup and construct two non-discreteHausdorff topologies turning I ∞ λ into a topological inverse semigroup.2. Algebraic properties of the semigroup I ∞ λ The definition of the semigroup I ∞ λ implies the following proposition: Proposition 2.1.
A partial map α ∈ I λ is an element of the semigroup I ∞ λ if and only if thefollowing assertions hold: ( i ) | λ \ dom α | = | λ \ ran α | ; and ( ii ) there exists a subset A ⊆ dom α ∩ ran α such that λ \ A is a finite subset of λ and the restriction α | A : A → A is the identity map. Proposition 2.2. ( i ) An element α of the semigroup I ∞ λ is an idempotent if and only if ( x ) α = x for every x ∈ dom α . ( ii ) If ε, ι ∈ E ( I ∞ λ ) , then ε ι if and only if dom ε ⊆ dom ι . ( iii ) The semilattice E ( I ∞ λ ) is isomorphic to ( P <ω ( λ ) , ∪ ) under the mapping ( ε ) h = λ \ dom ε . ( iv ) Every maximal chain in E ( I ∞ λ ) is an ω -chain. ( v ) α R β in I ∞ λ if and only if dom α = dom β . ( vi ) α L β in I ∞ λ if and only if ran α = ran β . ( vii ) α H β in I ∞ λ if and only if dom α = dom β and ran α = ran β . IVAN CHUCHMAN AND OLEG GUTIK ( viii ) α D β in I ∞ λ if and only if | λ \ dom α | = | λ \ dom β | . ( ix ) If n is a non-negative integer, then for every α, β ∈ I ∞ λ such that | λ \ dom α | = | λ \ dom β | = n there exist γ, δ ∈ I ∞ λ such that α = γ · β · δ and | λ \ dom γ | = | λ \ dom δ | = n . ( x ) For every non-negative integer n the set I n = { α ∈ I ∞ λ | | λ \ dom α | > n } is an ideal in I ∞ λ .Moreover, for every ideal I in I ∞ λ there exists an integer n > such that I is equal to I n . ( xi ) D = J in I ∞ λ . ( xii ) If λ and λ are infinite cardinals such that λ λ then I ∞ λ is a subsemigroup of the semigroup I ∞ λ . ( xiii ) ( I ∞ λ / J , ) is an ω -chain for any infinite cardinal λ .Proof. Statements ( i ) − ( iv ) are trivial and they follow from the definition of the semigroup I ∞ λ .( v ) Let be α, β ∈ I ∞ λ such that α R β . Since α I ∞ λ = β I ∞ λ and I ∞ λ is an inverse semigroup,Theorem 1.17 [5] implies that α I ∞ λ = αα − I ∞ λ , β I ∞ λ = ββ − I ∞ λ and hence αα − = ββ − . Thereforewe get that dom α = dom β .Conversely, let be α, β ∈ I ∞ λ such that dom α = dom β . Then αα − = ββ − . Since I ∞ λ is aninverse semigroup, Theorem 1.17 [5] implies that α I ∞ λ = αα − I ∞ λ = β I ∞ λ and hence α I ∞ λ = β I ∞ λ .The proof of statement ( vi ) is similar to ( v ).Statement ( vii ) follows from ( v ) and ( vi ).( viii ) Let α, β ∈ I ∞ λ be such that α D β . Then there exists γ ∈ I ∞ λ such that α L γ and γ R β .Therefore by statements ( v ) and ( vi ) we have that ran α = ran γ and dom γ = dom β . Then Propo-sition 2.1 implies that | λ \ ran γ | = | λ \ dom γ | and | λ \ ran β | = | λ \ dom β | , and hence we get that | λ \ dom α | = | λ \ dom β | .Let α and β are elements of the semigroup I ∞ λ such that | λ \ dom α | = | λ \ dom β | . Then Proposi-tion 2.1 implies that | λ \ ran α | = | λ \ dom α | and | λ \ ran β | = | λ \ dom β | . Let A α and A β be maximalsubsets of λ such that the sets λ \ A α and λ \ A β are finite and the restrictions α | A α : A α → A α and β | A β : A β → A β are identity maps. We put A = A α ∩ A β . Since λ \ A α and λ \ A β are finite subsets of λ we conclude that λ \ A is a finite subset of λ too. Since | λ \ dom α | = | λ \ dom β | < ω Proposition 2.1implies that | dom α \ A | = | ran α \ A | = | dom β \ A | = | ran β \ A | = n for some non-negative integer n . If n = 0, then α = β . Suppose that n >
1. Let { x , . . . , x n } = ran α \ A and { y , . . . , y n } = dom β \ A . We define γ = (cid:18) y · · · y n x · · · x n (cid:12)(cid:12)(cid:12)(cid:12) A (cid:19) . Then by statements ( v ) and ( vi ) we have that α L γ and γ R β in I ∞ λ . Hence α D β in I ∞ λ .( ix ) Let α and β be arbitrary elements of the semigroup I ∞ λ such that | λ \ dom α | = | λ \ dom β | = n for some non-negative integer n . Let A α and A β be maximal subsets of λ such that the sets λ \ A α and λ \ A β are finite and the restrictions α | A α : A α → A α and β | A β : A β → A β are identity maps.We put A = A α ∩ A β . Since λ \ A α and λ \ A β are finite subsets of λ we conclude that λ \ A is afinite subset of λ too. Since | λ \ dom α | = | λ \ dom β | the definition of the semigroup I ∞ λ impliesthat | dom α \ A | = | dom β \ A | < ω . If dom α \ A = dom β \ A = ∅ then α = β and hence α = γ · β · δ for γ = δ = I . Otherwise we put { x , . . . , x k } = dom α \ A , { y , . . . , y k } = dom β \ A , b = ( y ) β, . . . , b k = ( y k ) β and a = ( x ) α, . . . , a k = ( x k ) α , for some positive integer k . We define γ = (cid:18) x · · · x k y · · · y k (cid:12)(cid:12)(cid:12)(cid:12) A (cid:19) and δ = (cid:18) b · · · b k a · · · a k (cid:12)(cid:12)(cid:12)(cid:12) A (cid:19) . Then γ, δ ∈ I ∞ λ , | λ \ dom γ | = | λ \ dom δ | = n and α = γ · β · δ .( x ) Let α and β be arbitrary elements of the semigroup I ∞ λ . Since α and β are injective partialselfmaps almost everywhere the identity of the cardinal λ we conclude that | λ \ dom( α · β ) | > max {| λ \ dom α | , | λ \ dom β |} . This implies the first assertion of statement ( x ). N MONOIDS OF INJECTIVE PARTIAL SELFMAPS ALMOST EVERYWHERE THE IDENTITY 5
Let I be an ideal in I ∞ λ . Then the definition of the semigroup I ∞ λ implies that there exists α ∈ I such that | λ \ dom α | = min {| λ \ dom γ | | γ ∈ I } . Then | λ \ dom α | = n for some integer n >
0. Hence I ⊆ I n and by statement ( ix ) we get that I n ⊆ I .This implies the second assertion of the statement.Statement ( xi ) follows from statement ( ix ).( xii ) Let α = (cid:18) x · · · x n y · · · y n (cid:12)(cid:12)(cid:12)(cid:12) A (cid:19) be an arbitrary element of the semigroup I ∞ λ and B = λ \ λ .We put e α = (cid:18) x · · · x n y · · · y n (cid:12)(cid:12)(cid:12)(cid:12) A ∪ B (cid:19) . Obviously that e α ∈ I ∞ λ . Simple verifications show that the map h : I ∞ λ → I ∞ λ defined by theformula ( α ) h = e α is an isomorphic embedding of the semigroup I ∞ λ into I ∞ λ .Statement ( xiii ) follows from items ( viii ) and ( xi ). (cid:3) Later we shall need the following proposition:
Proposition 2.3.
Let λ be an arbitrary infinite cardinal. Then for every finite subset { x , . . . , x n } of λ the semigroups I ∞ λ and I ∞ η are isomorphic for η = λ \ { x , . . . , x n } .Proof. Since λ is infinite we conclude that there exists a bijective map f : λ → η . Then the bijection f generates a map h : I ∞ λ → I ∞ η such that the following condition holds:( α λ ) h = α η if and only if (( x ) f ) α η = (( x ) α λ ) f for every x ∈ λ, where α λ ∈ I ∞ λ and α η ∈ I ∞ η .Now we shall show that so defined map h is injective. Suppose to the contrary that there existdistinct elements α λ , β λ ∈ I ∞ λ such that ( α λ ) h = ( β λ ) h . We denote α η = ( α λ ) h and β η = ( β λ ) h .Then dom α η = dom β η and ran α η = ran β η and since f : λ → η is a bijective map we conclude thatdom α λ = dom β λ and ran α λ = ran β λ . Therefore there exists x ∈ ran α λ such that ( x ) α λ = ( x ) β λ .Since ( α λ ) h = ( β λ ) h we have that (( x ) f ) α η = (( x ) f ) β η . But (( x ) f ) α η = (( x ) α λ ) f and (( x ) f ) β η =(( x ) β λ ) f and since the map f : λ → η is bijective we conclude that ( x ) α λ = ( x ) β λ , a contradiction.The obtained contradiction implies that the map h : I ∞ λ → I ∞ η is injective.Let α η = (cid:18) x · · · x n y · · · y n (cid:12)(cid:12)(cid:12)(cid:12) A (cid:19) be an arbitrary element of the semigroup I ∞ η , where A ⊆ η and x , . . . , x n , y , . . . , y n ∈ η . Since themap f : λ → η is bijective we conclude that α λ = (cid:18) ( x ) f − · · · ( x n ) f − ( y ) f − · · · ( y n ) f − (cid:12)(cid:12)(cid:12)(cid:12) ( A ) f − (cid:19) is a partial bijective map from λ into λ such that the sets λ \ dom α λ and λ \ ran α λ are finite. Therefore α λ ∈ I ∞ λ and hence the map h : I ∞ λ → I ∞ η is bijective.Now we prove that the map h : I ∞ λ → I ∞ η is a homomorphism. We fix arbitrary elements α λ , β λ ∈ I ∞ λ and denote α η = ( α λ ) h and β η = ( β λ ) h . Then for every x ∈ ran α λ we have that (cid:0) ( x ) f (cid:1) ( α η · β η ) = (cid:16)(cid:0) ( x ) f (cid:1) α η (cid:17) β η = (cid:16)(cid:0) ( x ) α λ (cid:1) f (cid:17) β η = (cid:16)(cid:0) ( x ) α λ (cid:1) β λ (cid:17) f = (cid:0) ( x )( α λ · β λ ) (cid:1) f, and hence ( α λ · β λ ) h = α η · β η = ( α λ ) h · ( β λ ) h .Therefore h is an isomorphism from the semigroup I ∞ λ onto I ∞ η . (cid:3) Proposition 2.4.
Let λ be an arbitrary infinite cardinal. Then for every idempotent ε of the semigroup I ∞ λ the semigroups I ∞ λ ( ε ) = ε · I ∞ λ · ε and I ∞ λ are isomorphic. IVAN CHUCHMAN AND OLEG GUTIK
Proof.
Since I ∞ λ ( ε ) = ε · I ∞ λ · ε = ε · I ∞ λ ∩ I ∞ λ · ε == { α ∈ I ∞ λ | dom α ⊆ dom ε } ∩ { α ∈ I ∞ λ | ran α ⊆ ran ε } == { α ∈ I ∞ λ | dom α ⊆ dom ε and ran α ⊆ ran ε } , Proposition 2.3 implies the assertion of the proposition. (cid:3)
Proposition 2.5.
For every α, β ∈ I ∞ λ , both sets { χ ∈ I ∞ λ | α · χ = β } and { χ ∈ I ∞ λ | χ · α = β } arefinite. Consequently, every right translation and every left translation by an element of the semigroup I ∞ λ is a finite-to-one map.Proof. We denote S = { χ ∈ I ∞ λ | α · χ = β } and T = { χ ∈ I ∞ λ | α − · α · χ = α − · β } . Then S ⊆ T and the restriction of any partial map χ ∈ T to dom( α − · α ) coincides with the partialmap α − · β . Since every partial map from the semigroup I ∞ λ is an injective partial selfmap almosteverywhere the identity we have that there exist maximal subsets A α − α and A α − β in λ such thatthe sets λ \ A α − α and λ \ A α − β are finite and the restrictions ( α − · α ) | A α − α : A α − α → A α − α and( α − · β ) | A α − β : A α − β → A α − β are identity maps. We put A = A α − α ∩ A α − β . Then the definitionof the semigroup I ∞ λ implies that the restrictions ( α − · α ) | A : A → A and ( α − · β ) | A : A → A areidentity maps and the set λ \ A is finite. This implies that the set T is finite and hence the set S isfinite too. (cid:3) For an arbitrary non-zero cardinal λ we denote by S ∞ ( λ ) the group of all bijective transformationsof λ with finite supports (i.e., α ∈ S ∞ ( λ ) if and only if the set { x ∈ λ | ( x ) α = x } is finite).The definition of the semigroup I ∞ λ and Proposition 2.4 imply the following proposition: Proposition 2.6.
Every maximal subgroup of the semigroup I ∞ λ is isomorphic to S ∞ ( λ ) . On congruences on the semigroup I ∞ λ If R is an arbitrary congruence on a semigroup S , then we denote by Φ R : S → S/ R the naturalhomomorphisms from S onto S/ R . Also we denote by Ω S and ∆ S the universal and the identity congruences, respectively, on the semigroup S , i. e., Ω( S ) = S × S and ∆( S ) = { ( s, s ) | s ∈ S } .The following lemma follows from the definition of a congruence on a semilattice: Lemma 3.1.
Let R is an arbitrary congruence on a semilattice E . Let a and b be elements of thesemilattice E such that a R b . Then ( i ) a R ( ab ) ; and ( ii ) if a b then a R c for all c ∈ E such that a c b . Proposition 3.2.
Let R be an arbitrary congruence on the semigroup I ∞ λ . Let ε and ϕ be idempotentsof I ∞ λ such that ε R ϕ and ε ϕ . If | dom ϕ \ dom ε | = 1 then the following conditions hold: ( i ) ϕ R ι for all idempotents ι ∈ ↓ ϕ ; and ( ii ) ϕ R χ for all idempotents χ ∈ I ∞ λ such that | λ \ dom ϕ | = | λ \ dom χ | .Proof. ( i ) First we shall show that ϕ R ψ for all idempotents ψ ∈ ↓ ε . By Proposition 2.2 ( iv ) there existsa maximal (not necessary unique) ω -chain L in E ( I ∞ λ ) which contains ε and ψ . Let L = { ε , . . . , ε n } be a maximal subchain in L such that ψ = ε n < . . . < ε = ε , where n is some positive integer. Theexistence of the subchain L follows from Proposition 2.2 ( iv ) too. Let x n = dom ε n − \ dom ε n , x n − = dom ε n − \ dom ε n − , . . . , x = dom ε \ dom ε , x = dom ϕ \ dom ε . We put α = (cid:18) x x (cid:12)(cid:12)(cid:12)(cid:12) dom ε (cid:19) , α = (cid:18) x x (cid:12)(cid:12)(cid:12)(cid:12) dom ε (cid:19) , . . . , α n − = (cid:18) x n − x n (cid:12)(cid:12)(cid:12)(cid:12) dom ε n (cid:19) . N MONOIDS OF INJECTIVE PARTIAL SELFMAPS ALMOST EVERYWHERE THE IDENTITY 7
Then we have that α − · ϕ · α = ε and α − · ε · α = ε ; α − · ε · α = ε and α − · ε · α = ε ; · · · · · · · · · α − n − · ε n − · α n − = ε n − and α − n − · ε n − · α n − = ε n , and hence ε R ε , ε R ε , . . . , ε n − R ε n . Since ϕ R ε we have that ϕ R ε n . This completes the proof of thestatement.Let ι be an arbitrary idempotent of the semigroup I ∞ λ such that ι ∈ ↓ ϕ . We put ι = ε · ι . Thenby previous part of the proof we have that ι R ϕ and hence by Lemma 3.1 we get ι R ϕ .( ii ) Let χ be an arbitrary idempotent of the semigroup I ∞ λ such that ϕ = χ and | λ \ dom ϕ | = | λ \ dom χ | . Then ε · χ ϕ and hence by statement ( i ) we get that ( ε · χ ) R ϕ . Since | λ \ dom ϕ | = | λ \ dom χ | we conclude that | dom ϕ \ dom( ε · χ ) | = | dom χ \ dom( ε · χ ) | . Let be { x , . . . , x k } = dom ϕ \ dom( ε · χ )and { y , . . . , y k } = dom χ \ dom( ε · χ ). We put α = (cid:18) x · · · x k y · · · y k (cid:12)(cid:12)(cid:12)(cid:12) dom( ε · χ ) (cid:19) . Then α − · ϕ · α = χ and α − · ( ε · χ ) · α = ε · χ . Therefore we get that ( ε · χ ) R χ and hence ϕ R χ . Thiscompletes the proof of our statement. (cid:3) Theorem 3.3.
Let R be an arbitrary congruence on the semigroup I ∞ λ and ε and ϕ be distinct R -equivalent idempotents of I ∞ λ . Then α R ε for every α ∈ I ∞ λ such that | λ \ dom α | > min {| λ \ dom ϕ | , | λ \ dom ε |} . Proof.
In the case when α is an idempotent of the semigroup I ∞ λ the statement of the theorem followsfrom Lemma 3.1 and Proposition 3.2.Suppose that α is an arbitrary non-idempotent element of the semigroup I ∞ λ such that | λ \ dom α | > max {| λ \ dom ϕ | , | λ \ dom ε |} . Since I ∞ λ is an inverse semigroup we have that α · α − · α = α andPropositions 2.1 and 2.2 imply that | λ \ dom α | = | λ \ dom α − | = | λ \ dom( α · α − ) | = | λ \ dom( α − · α ) | > min {| λ \ dom ϕ | , | λ \ dom ε |} . Hence ( α · α − ) R ε and by Proposition 3.2 we have that ( α · α − ) R ι for every idempotent ι of thesemigroup I ∞ λ such that ι ∈ ↓ ε . Definition of the semigroup I ∞ λ implies that for every α ∈ I ∞ λ thereexists an idempotent ς α ∈ I ∞ λ such that α · ς = ς · α = ς · ( α · α − ) = ς for all idempotents ς ∈ I ∞ λ such that ς ∈ ↓ ς α . Let ν = ς α · ε . Then ( α · α − ) R ν and α · ν = ν · α = ν · ( α · α − ) = ν . Therefore weget ( α )Φ R = ( α · α − · α )Φ R = ( α · α − )Φ R · ( α )Φ R = ( ν )Φ R · ( α )Φ R = ( ν · α )Φ R = ( ν )Φ R and α R ν . Hence we have that α R ε . (cid:3) Proposition 3.4.
Let R be an arbitrary congruence on the semigroup I ∞ λ . Let ε be an idempotentof I ∞ λ such that | λ \ dom ε | > and the following conditions hold: ( i ) there exists an idempotent ϕ ∈ I ∞ λ such that ε R ϕ and | λ \ dom ϕ | > | λ \ dom ε | ; and ( ii ) does not exist an idempotent ψ ∈ I ∞ λ such that ε R ψ and | λ \ dom ψ | < | λ \ dom ε | .Then there exists no element α of the semigroup I ∞ λ such that ε R α and | λ \ dom α | < | λ \ dom ε | .Proof. Suppose to the contrary that there exists α ∈ I ∞ λ such that ε R α and | λ \ dom α | < | λ \ dom ε | .Since I ∞ λ is an inverse semigroup Lemma III.1.1 [21] implies that ε R α − and hence ε R ( α · α − ). But | λ \ dom( α · α − ) | = | λ \ dom α | < | λ \ dom ε | , a contradiction. An obtained contradiction implies thestatement of the proposition. (cid:3) Proposition 3.5.
Let R be an arbitrary congruence on the semigroup I ∞ λ . Let α and β be non- H -equivalent elements of I ∞ λ such that α R β . Then γ R α for all γ ∈ I ∞ λ such that | λ \ dom γ | > min {| λ \ dom α | , | λ \ dom β |} . IVAN CHUCHMAN AND OLEG GUTIK
Proof.
Since α and β are non- H -equivalent elements of the inverse semigroup I ∞ λ we conclude thatat least one of the following conditions holds:( i ) α · α − = β · β − ;( ii ) α − · α = β − · β .Suppose that the case α · α − = β · β − holds. In the other case the proof is similar. Since I ∞ λ is aninverse semigroup Lemma III.1.1 [21] implies that β − R α − and hence ( β · β − ) R ( α · α − ). Then wehave that | λ \ dom α | = | λ \ dom( α · α − ) | and | λ \ dom β | = | λ \ dom( β · β − ) | and hence the assumptions of the Theorem 3.3 hold. This completes the proof of the proposition. (cid:3) Proposition 3.6.
Let R be an arbitrary congruence on the semigroup I ∞ λ . If α and β are distinct H -equivalent elements of I ∞ λ such that α R β , then γ R α for all γ ∈ I ∞ λ such that | λ \ dom γ | > | λ \ dom α | . Proof.
Since I ∞ λ is an inverse semigroup Theorem 2.20 [5] and Proposition 2.2 ( viii ) imply thatwithout loss of generality we can assume that α and β are elements of a maximal subgroup H ( ε ) of I ∞ λ with unity ε . Since ( α · α − ) R ( β · α − ) we can assume that α is an identity of the subgroup H ( ε ).Let x ∈ dom α such that ( x ) β = x . We put ε : dom α \ { x } → dom α \ { x } be an identity map. Then ε · α = ε and ran( ε · β ) = ran( ε ). Therefore by Proposition 2.2 ( vii ) we get that the elements ε and ε · β are not H -equivalent. Since | λ \ dom ε | = | λ \ dom( ε · β ) | we have that the assumptionsof Proposition 3.5 hold. This completes the proof of the proposition. (cid:3) Theorem 3.3 and Propositions 3.4, 3.5 and 3.6 imply the following proposition:
Proposition 3.7.
Let R be an arbitrary congruence on the semigroup I ∞ λ . Let α and β be distinct H -equivalent elements of I ∞ λ such that α R β and suppose that there does not exist γ ∈ I ∞ λ such that α R γ and | λ \ dom γ | < | λ \ dom α | . Then elements µ, ν ∈ I ∞ λ with | λ \ dom µ | < | λ \ dom α | and | λ \ dom ν | < | λ \ dom α | are R -equivalent if and only if µ = ν . Definition 3.8.
For every non-negative integer n we denote by K n ( I ) the congruence on the semigroup I ∞ λ generated by the ideal I n , i. e., K n ( I ) = ( I n × I n ) ∪ ∆( I ∞ λ ). We observe that K ( I ) = Ω( I ∞ λ ). Remark 3.9.
The group S ∞ ( λ ) has only one non-trivial normal subgroup: that is a group A ∞ ( λ ) ofall even permutations of the set λ (see [10, pp. 313–314, Example] or [18]). Therefore every non-trivialhomomorphism of S ∞ ( λ ) is either an isomorphism or its image is a two-elements cyclic group. Definition 3.10.
Fix an arbitrary non-negative integer n . We shall say that elements α and β of thesemigroup I ∞ λ are n S ∞ -equivalent if the following conditions hold:( i ) α H β ; and( ii ) | λ \ dom α | = | λ \ dom β | = n .We define a relation K n ( S ∞ ) on the semigroup I ∞ λ as follows: K n ( S ∞ ) = { ( α, β ) | ( α, β ) ∈ n S ∞ } ∪ ( I n +1 × I n +1 ) ∪ ∆( I ∞ λ ) . Simple verifications show that so defined relation K n ( S ∞ ) on I ∞ λ is an equivalence relation for everynon-negative integer n . Proposition 3.11.
The relation K n ( S ∞ ) is a congruence on the semigroup I ∞ λ .Proof. First we consider the case when n = 0. If α and β are distinct elements of the semigroup I ∞ λ such that α K ( S ∞ ) β , then either α, β ∈ H ( I ) or α, β ∈ I . Suppose that α, β ∈ H ( I ). Thenfor every γ ∈ I ∞ λ we have that either α · γ, β · γ ∈ H ( I ) or α · γ, β · γ ∈ I , and similarly we getthat either γ · α, γ · β ∈ H ( I ) or γ · α, γ · β ∈ I . If α, β ∈ I then for every γ ∈ I ∞ λ we have that α · γ, β · γ, α · γ, β · γ ∈ I . Therefore K ( S ∞ ) is a congruence on the semigroup I ∞ λ .Suppose that n is an arbitrary positive integer. Let α and β be distinct elements of the semigroup I ∞ λ such that α K n ( S ∞ ) β . The definition of the relation K n ( S ∞ ) implies that only one of the followingconditions holds: N MONOIDS OF INJECTIVE PARTIAL SELFMAPS ALMOST EVERYWHERE THE IDENTITY 9 ( i ) | λ \ dom α | = | λ \ dom β | = n ; or( ii ) | λ \ dom α | > n and | λ \ dom β | > n .First we suppose that | λ \ dom α | = | λ \ dom β | = n . Let γ be an arbitrary element of the semigroup I ∞ λ . We consider two cases: a ) dom α ⊆ ran γ ; and b ) dom α * ran γ .Since the elements α and β are H -equivalent in I ∞ λ Proposition 2.2 ( vii ) implies that in case a ) wehave that dom( γ · α ) = dom( γ · β ) and ran( γ · α ) = ran( γ · β ). Then again by Proposition 2.2 ( vii ) theelements γ · α and γ · β are H -equivalent in I ∞ λ . Since dom α ⊆ ran γ we get that | λ \ dom( γ · α ) | = | λ \ dom( γ · β ) | = n . Hence we obtain that ( γ · α ) K n ( S ∞ )( γ · β ). In case b ) we have that γ · α, γ · β ∈ I n +1 and hence ( γ · α ) K n ( S ∞ )( γ · β ).The proof the assertion that α K n ( S ∞ ) β implies ( α · δ ) K n ( S ∞ )( β · δ ) for every δ ∈ I ∞ λ is similar.Suppose that | λ \ dom α | > n and | λ \ dom β | > n . Then α, β ∈ I n +1 . By Proposition 2.2 ( x ) wehave that γ · α, γ · β, α · δ, β · δ ∈ I n +1 and hence ( γ · α ) K n ( S ∞ )( γ · β ) and ( α · δ ) K n ( S ∞ )( β · δ ) for all γ, δ ∈ I ∞ λ . This completes the proof of the proposition. (cid:3) Definition 3.12.
Fix an arbitrary non-negative integer n . We shall say that elements α and β of thesemigroup I ∞ λ are n A ∞ -equivalent if the following conditions hold:( i ) α H β ;( ii ) α · β − is an even permutation of the set dom α ; and( iii ) | λ \ dom α | = | λ \ dom β | = n .We define a relation K n ( A ∞ ) on the semigroup I ∞ λ as follows: K n ( A ∞ ) = { ( α, β ) | ( α, β ) ∈ n A ∞ } ∪ ( I n +1 × I n +1 ) ∪ ∆( I ∞ λ ) . Simple verifications show that so defined relation K n ( A ∞ ) on I ∞ λ is an equivalence relation for everynon-negative integer n . Proposition 3.13.
The relation K n ( A ∞ ) is a congruence on the semigroup I ∞ λ .Proof. First we consider the case when n = 0. If α and β are distinct elements of the semigroup I ∞ λ such that α K ( S ∞ ) β , then either α, β ∈ H ( I ) or α, β ∈ I . Suppose that α, β ∈ H ( I ). Then for every γ ∈ H ( I ) we have that α · γ, β · γ, γ · α, γ · β ∈ H ( I ). Then ( α · γ ) · ( β · γ ) − = α · γ · γ − · β − = α · β − is an even permutation of the set λ . Also, since α · β − is an even permutation of the set λ we get that( γ · α ) · ( γ · β ) − = γ · α · β − · γ − is an even permutation of the set λ too. For every γ ∈ I we have that α · γ, β · γ, γ · α, γ · β ∈ I . If α, β ∈ I then for every γ ∈ I ∞ λ we have that α · γ, β · γ, α · γ, β · γ ∈ I .Therefore K ( A ∞ ) is a congruence on the semigroup I ∞ λ .Suppose that n is an arbitrary positive integer. Let α and β be distinct elements of the semigroup I ∞ λ such that α K n ( A ∞ ) β . The definition of the relation K n ( A ∞ ) implies that only one of the followingconditions holds:( i ) | λ \ dom α | = | λ \ dom β | = n ; or( ii ) | λ \ dom α | > n and | λ \ dom β | > n .First we suppose that | λ \ dom α | = | λ \ dom β | = n . Let γ be an arbitrary element of the semigroup I ∞ λ . We consider two cases: a ) dom α ⊆ ran γ ; and b ) dom α * ran γ .Suppose case a ) holds. Since the elements α and β are H -equivalent in I ∞ λ we have that Propo-sition 2.2 ( vii ) implies that dom( γ · α ) = dom( γ · β ) and ran( γ · α ) = ran( γ · β ). Then again byProposition 2.2 ( vii ) the elements γ · α and γ · β are H -equivalent in I ∞ λ . Since dom α ⊆ ran γ we get that | λ \ dom( γ · α ) | = | λ \ dom( γ · β ) | = n . We define a partial map γ : λ ⇀ λ as follows γ = γ | (dom α ) γ − : (dom α ) γ − → dom α . Then we get that | λ \ dom γ | = | λ \ dom α | = | λ \ dom β | = n , γ · α = γ · α , γ · β = γ · β and hence ( γ · α ) · ( γ · β ) − = ( γ · α ) · ( γ · β ) − = γ · α · β − · γ − .Since α · β − is an even permutation of the set dom α we conclude that γ · α · β − · γ − is an even permutation of the set dom γ = (dom α ) γ − . Hence we obtain that ( γ · α ) K n ( A ∞ )( γ · β ). In case b )we have that γ · α, γ · β ∈ I n +1 and hence ( γ · α ) K n ( A ∞ )( γ · β ).The proof the assertion that α K n ( A ∞ ) β implies ( α · δ ) K n ( A ∞ )( β · δ ) for every δ ∈ I ∞ λ is similar.Suppose that | λ \ dom α | > n and | λ \ dom β | > n . Then α, β ∈ I n +1 . By Proposition 2.2 ( x ) wehave that γ · α, γ · β, α · δ, β · δ ∈ I n +1 and hence ( γ · α ) K n ( A ∞ )( γ · β ) and ( α · δ ) K n ( A ∞ )( β · δ ), for all γ, δ ∈ I ∞ λ . This completes the proof of the proposition. (cid:3) Theorem 3.14.
The family
Cong ( I ∞ λ ) = { ∆( I ∞ λ ) , Ω( I ∞ λ ) } ∪ { K n ( S ∞ ) | n = 0 , , , . . . } ∪ { K n ( A ∞ ) | n = 0 , , , . . . }∪∪ { K n ( I n ) | n = 1 , , . . . } determines all congruences on the semigroup I ∞ λ .Proof. Let R be non-identity congruence on the semigroup I ∞ λ . Since the set of all non-negativeintegers with respect to the usual order is well ordered there exists a minimal non-negative integer n such that there are two distinct elements α and β in I ∞ λ such that α R β andmin {| λ \ dom α | , | λ \ dom β |} = n, i.e., for any non-negative integer m < n if for α and β in I ∞ λ such that α R β andmin {| λ \ dom α | , | λ \ dom β |} = m then α = β .We consider two cases:( i ) | λ \ dom α | 6 = | λ \ dom β | ; and( ii ) | λ \ dom α | = | λ \ dom β | .Suppose case ( i ) holds and | λ \ dom α | = n < | λ \ dom β | . Then α and β are not H -equivalentelements in I ∞ λ and hence by Proposition 3.5 we obtain that α R γ for all γ ∈ I ∞ λ with | λ \ dom γ | > n .Then Proposition 3.7 implies that µ R ν if and only if µ = ν for all elements µ, ν ∈ I ∞ λ such that | λ \ dom µ | < n and | λ \ dom ν | < n . Hence we get that R = K n ( I ). We observe if n = 0 then R = Ω( I ∞ λ ).We henceforth assume that case ( ii ) holds.If α and β are not H -equivalent elements in I ∞ λ and then by Proposition 3.5 we have that α R γ for all γ ∈ I ∞ λ such that | λ \ dom γ | > n . Then Proposition 3.7 implies that µ R ν if and only if µ = ν for all elements µ, ν ∈ I ∞ λ such that | λ \ dom µ | < n and | λ \ dom ν | < n , and hence we have that R = K n ( I ). Also in this case if n = 0 then R = Ω( I ∞ λ ).Suppose that α and β are H -equivalent elements in I ∞ λ and there exists no non- H -equivalentelement δ of the semigroup I ∞ λ such that α R δ . Otherwise by the previous part of the proof we havethat R = K n ( I ). Since ( α · α − ) R ( β · α − ) we conclude that without loss of generality we can assumethat α is an identity element of H -class H ( α ) which contains α and β = α . Since α is an idempotentof the semigroup I ∞ λ we have that dom α = ran α and the restriction α | dom α : dom α → dom α is anidentity map. Also we observe that the restriction of the partial map β | dom α : dom α → dom α is apermutation of the set dom α . Therefore without loss of generality we can consider β as a permutationof the set dom α .We consider two cases:(1) β is an odd permutation of the set dom α ; and(2) β is an even permutation of the set dom α .Suppose that β is an odd permutation of the set dom α . Since H ( α ) is a subgroup of the semigroup I ∞ λ we conclude that the image ( H ( α ))Φ R of H ( α ) is a subgroup in I ∞ λ / R . Since the subgroup H ( α ) is isomorphic to the group S ∞ ( λ ) and the group of all even permutations A ∞ ( λ ) of the set λ is a unique normal subgroup in S ∞ ( λ ) (see [10, pp. 313–314, Example] or [18]) we conclude that theimage ( H ( α ))Φ R is singleton. Then by Theorem 2.20 [5] and Proposition 2.2 ( viii ) for every γ ∈ I ∞ λ N MONOIDS OF INJECTIVE PARTIAL SELFMAPS ALMOST EVERYWHERE THE IDENTITY 11 with | λ \ dom γ | = | λ \ dom α | the image ( H γ )Φ R of the H -class H γ which contains the element γ issingleton and hence by Propositions 3.5, 3.6 and 3.7 we get that R = K n ( S ∞ ).Suppose that β is an even permutation of the set dom α . If the subgroup H ( α ) contains an oddpermutation δ of the set dom α then by previous proof we get that R = K n ( S ∞ ). Suppose the subgroup H ( α ) does not contain an odd permutation δ of the set dom α . Since the subgroup H ( α ) is isomorphicto the group S ∞ ( λ ) and the group of all even permutations A ∞ ( λ ) of the set λ is a unique normalsubgroup in S ∞ ( λ ) we conclude that the image ( H ( α ))Φ R is a two-element subgroup in I ∞ λ / R . Thenby Theorem 2.20 [5] and Proposition 2.2 ( viii ) for every γ ∈ I ∞ λ with | λ \ dom γ | = | λ \ dom α | theimage ( H γ )Φ R of the H -class H γ which contains the element γ is a two-element subset in I ∞ λ / R andhence by Propositions 3.5, 3.6 and 3.7 we get that R = K n ( A ∞ ). (cid:3) On topologizations of the free semilattice ( P <ω ( λ ) , ∪ ) Definition 4.1 ([4]) . We shall say that a semigroup S has the F - property if for every a, b, c, d ∈ S the sets { x ∈ S | a · x = b } and { x ∈ S | x · c = d } are finite or empty.Recall [9] an element x of a semitopological semilattice S is a local minimum if there exists an openneighbourhood U ( x ) of x such that U ( x ) ∩ ↓ x = { x } . This is equivalent to statement that ↑ x is anopen subset in S .A topological space X is called Baire if for each sequence A , A , . . . , A i , . . . of nowhere dense subsetsof X the union ∞ [ i =1 A i is a co-dense subset of X [7]. A Tychonoff space X is called ˇCech complete if forevery compactification cX of X the remainder cX \ c ( X ) is an F σ -set in cX [7].A topological space X is called hereditary Baire if every closed subset of X is a Baire space [7].Every ˇCech complete (and hence locally compact) space is hereditary Baire (see [7, Theorem 3.9.6]).We shall say that a Hausdorff semitopological semigroup S is an I - Baire space if for every s ∈ S either sS or Ss is a Baire space [4]. Remark 4.2.
We observe that every left ideal Ss and every right ideal sS of a regular semigroup S is generated by its idempotents. Therefore every principal left (right) ideal of a regular Hausdorffsemitopological semigroup S is a closed subset of S . Hence every regular Hausdorff hereditary Bairesemitopological semigroup is a I -Baire space. Theorem 4.3.
Let S be a semilattice with the F -property. Then every I -Baire topology τ on S suchthat ( S, τ ) is a Hausdorff semitopological semilattice is discrete.Proof. Let x be an arbitrary element of the semilattice S . We need to show that x is an isolated pointin ( S, τ ).Since τ is an I -Baire topology on S we conclude that the subspace ↓ x is Baire. We denote S x = ↓ x .For every positive integer n we put F n = { y ∈ S x | |↑ y | = n } . Then we have that S x = S ∞ i =1 F n . Since the topological space S x is Baire we conclude that that thereexists F n ∈ F such that Int S x ( F n ) = ∅ . We fix an arbitrary y ∈ Int S x ( F n ). We observe that thedefinition of the family { F n | n ∈ N } implies that for every non-empty subset F n and for any s ∈ F n the sets ↑ s ∩ F n and ↓ s ∩ F n are singleton. This implies that y is a local minimum in S x , i.e., ↑ y isan open subset of S . Since the semilattice S x has the F -property we conclude that the Hausdorffnessof S implies that x is an isolated point in S x . Then x is a local minimum in S and hence ↑ x is anopen subset in S . Since the semilattice S has the F -property we conclude that the Hausdorffness of S implies that x is an isolated point in S . (cid:3) Remark 4.4.
We observe that the statement of Theorem 4.3 is true for a T -semitopological I -Bairesemilattice with the F -property.Since every ˇCech complete (and hence locally compact) space is hereditary Baire, Theorem 4.3implies the following corollary: Corollary 4.5.
Let S be a semilattice with the F -property. Then every ˇCech complete (locally compact)topology τ on S such that ( S, τ ) is a semitopological semilattice is discrete. Since the free semilattice ( P <ω ( λ ) , ∪ ) has F -property, Theorem 4.3 implies the following corollary: Corollary 4.6.
Every Hausdorff I -Baire ( ˇCech complete, locally compact) topology τ on the freesemilattice P <ω ( λ ) such that ( P <ω ( λ ) , τ ) is a semitopological semilattice is discrete. On a topological semigroup I ∞ λ Theorem 5.1.
Every hereditary Baire topology τ on the semigroup I ∞ ω such that ( I ∞ ω , τ ) is a Haus-dorff semitopological semigroup is discrete.Proof. Let α be an arbitrary element of the the semigroup I ∞ ω . We need to show that α is an isolatedpoint in ( I ∞ ω , τ ).For every non-negative integer n we denote C n = I ∞ ω \ I n +1 .By induction we shall prove that for every non-negative integer n the following statement holds: every α ∈ C n is an isolated point in ( I ∞ ω , τ ).First we shall show that our statement is true for n = 0. We define a family C = {{ β } | β ∈ I ∞ ω } .Since the topological space ( I ∞ ω , τ ) is Baire we have that the family C has an element with non-empty interior and hence the topological space ( I ∞ ω , τ ) has an isolated point γ in ( I ∞ ω , τ ). Then | ω \ dom α | = 0 and hence statements ( viii ) − ( xi ) of Proposition 2.2 imply that there exist µ, ν ∈ I ∞ ω such that µ · α · ν = γ . Since translations in ( I ∞ ω , τ ) are continuous we conclude that Hausdorffnessof the space ( I ∞ ω , τ ) and Proposition 2.5 imply that α an isolated point in ( I ∞ ω , τ ).Suppose our statement is true for all n < k , k ∈ N . We shall show that its is true for n = k . Ourassumption implies that I k is a closed subset of ( I ∞ ω , τ ). Later we shall denote by τ k the topologyinduced from ( I ∞ ω , τ ) onto I k . Then ( I k , τ k ) is a Baire space. We define a family C k = {{ β } | β ∈ I k } .Since the topological space ( I k , τ k ) is Baire we have that the family C k has an element with non-empty interior and hence the topological space ( I k , τ k ) has an isolated point γ in ( I k , τ k ). Let U ( γ )be an open neighbourhood U ( γ ) of γ in ( I ∞ ω , τ ) such that U ( γ ) ∩ I k = { γ } . Since ( I ∞ ω , τ ) is asemitopological semigroup we have that there exists an open neighbourhood V ( γ ) of γ in ( I ∞ ω , τ )such that V ( γ ) ⊆ U ( γ ) and γ · γ − · V ( γ ) ⊆ U ( γ ). We remark that γ · γ − · V ( γ ) ⊆ { γ } . Hence byProposition 2.5 the neighbourhood V ( γ ) is finite and Hausdorffness of the space ( I ∞ ω , τ ) implies that γ an isolated point in ( I ∞ ω , τ ). Let α be an arbitrary element of the set I k \ I k +1 . Then | ω \ dom α | = k and hence statements ( viii ) − ( xi ) of Proposition 2.2 imply that there exist µ, ν ∈ I ∞ ω such that µ · α · ν = γ . Since translations in ( I ∞ ω , τ ) are continuous we conclude that Hausdorffness of the space( I ∞ ω , τ ) and Proposition 2.5 imply that α an isolated point in ( I ∞ ω , τ ). This completes the proof ofour theorem. (cid:3) Remark 5.2.
We observe that the statement of Theorem 5.1 holds for every topology τ on thesemigroup I ∞ ω such that ( I ∞ ω , τ ) is a Hausdorff semitopological semigroup and every (two-sided)ideal in ( I ∞ ω , τ ) is a Baire space.Theorem 5.1 implies the following corollary: Corollary 5.3.
Every ˇCech complete (locally compact) topology τ on the semigroup I ∞ ω such that ( I ∞ ω , τ ) is a Hausdorff semitopological semigroup is discrete. Theorem 5.4.
Let λ be an infinite cardinal and S be a topological semigroup which contains a densediscrete subsemigroup I ∞ λ . If I = S \ I ∞ λ = ∅ then I is an ideal of S .Proof. Suppose that I is not an ideal of S . Then at least one of the following conditions holds:1) I · I ∞ λ * I, I ∞ λ · I * I, or 3) I · I * I. Since I ∞ λ is a discrete dense subspace of S , Theorem 3.5.8 [7] implies that I ∞ λ is an open subspaceof S . Suppose there exist a ∈ I ∞ λ and b ∈ I such that b · a = c / ∈ I . Since I ∞ λ is a dense opendiscrete subspace of S the continuity of the semigroup operation in S implies that there exists an open N MONOIDS OF INJECTIVE PARTIAL SELFMAPS ALMOST EVERYWHERE THE IDENTITY 13 neighbourhood U ( b ) of b in S such that U ( b ) · { a } = { c } . But by Proposition 2.5 the equation x · a = c has finitely many solutions in I ∞ λ . This contradicts the assumption that b ∈ S \ I ∞ λ . Therefore b · a = c ∈ I and hence I · I ∞ λ ⊆ I . The proof of the inclusion I ∞ λ · I ⊆ I is similar.Suppose there exist a, b ∈ I such that a · b = c / ∈ I . Since I ∞ λ is a dense open discrete subspace of S the continuity of the semigroup operation in S implies that there exist open neighbourhoods U ( a ) and U ( b ) of a and b in S , respectively, such that U ( a ) · U ( b ) = { c } . But by Proposition 2.5 the equations x · b = c and a · y = c have finitely many solutions in I ∞ λ . This contradicts the assumption that a, b ∈ S \ I ∞ λ . Therefore a · b = c ∈ I and hence I · I ⊆ I . (cid:3) Proposition 5.5.
Let S be a topological semigroup which contains a dense discrete subsemigroup I ∞ λ .Then for every c ∈ I ∞ λ the set D c ( I ∞ λ ) = { ( x, y ) ∈ I ∞ λ × I ∞ λ | x · y = c } is a closed-and-open subset of S × S .Proof. Since I ∞ λ is a discrete subspace of S we have that D c ( I ∞ λ ) is an open subset of S × S .Suppose that there exists c ∈ I ∞ λ such that D c ( I ∞ λ ) is a non-closed subset of S × S . Then thereexists an accumulation point ( a, b ) ∈ S × S of the set D c ( I ∞ λ ). The continuity of the semigroupoperation in S implies that a · b = c . But I ∞ λ × I ∞ λ is a discrete subspace of S × S and hence byTheorem 5.4 the points a and b belong to the ideal I = S \ I ∞ λ and hence a · b ∈ S \ I ∞ λ cannot beequal to c . (cid:3) A topological space X is defined to be pseudocompact if each locally finite open cover of X is finite.According to [7, Theorem 3.10.22] a Tychonoff topological space X is pseudocompact if and only ifeach continuous real-valued function on X is bounded. Theorem 5.6.
If a topological semigroup S contains I ∞ λ as a dense discrete subsemigroup then thesquare S × S is not pseudocompact.Proof. Since the square S × S contains an infinite closed-and-open discrete subspace D c ( I ∞ λ ), weconclude that S × S fails to be pseudocompact (see [7, Ex. 3.10.F(d)] or [6]). (cid:3) A topological space X is called countably compact if any countable open cover of X contains a finitesubcover [7]. We observe that every Hausdorff countably compact space is pseudocompact.Since the closure of an arbitrary subspace of a countably compact space is countably compact (see[7, Theorem 3.10.4]) Theorem 5.6 implies the following corollary: Corollary 5.7.
For every infinite cardinal λ the discrete semigroup I ∞ λ does not embed into a topo-logical semigroup S with the countably compact square S × S . Since every compact topological space is countably compact Theorem 3.24 [7] and Corollary 5.7imply
Corollary 5.8.
For every infinite cardinal λ the discrete semigroup I ∞ λ does not embed into a compacttopological semigroup. We recall that the Stone- ˇCech compactification of a Tychonoff space X is a compact Hausdorff space βX containing X as a dense subspace so that each continuous map f : X → Y to a compact Hausdorffspace Y extends to a continuous map f : βX → Y [7]. Theorem 5.9.
For every infinite cardinal λ the discrete semigroup I ∞ λ does not embed into a Ty-chonoff topological semigroup S with the pseudocompact square S × S .Proof. By Theorem 1.3 [1] for any topological semigroup S with the pseudocompact square S × S thesemigroup operation µ : S × S → S extends to a continuous semigroup operation βµ : βS × βS → βS ,so S is a subsemigroup of the compact topological semigroup βS . Then Corollary 5.8 implies thestatement of the theorem. (cid:3) The following example shows that there exists a non-discrete topology τ F on the semigroup I ∞ λ such that ( I ∞ λ , τ F ) is a Tychonoff topological inverse semigroup. Example 5.10.
We define a topology τ F on the semigroup I ∞ λ as follows. For every α ∈ I ∞ λ wedefine a family B F ( α ) = { U α ( F ) | F is a finite subset of dom α } , where U α ( F ) = { β ∈ I ∞ λ | dom α = dom β, ran α = ran β and ( x ) β = ( x ) α for all x ∈ F } . Since conditions (BP1)–(BP3) [7] hold for the family { B F ( α ) } α ∈ I ∞ λ we conclude that the family { B F ( α ) } α ∈ I ∞ λ is the base of the topology τ F on the semigroup I ∞ λ . Proposition 5.11. ( I ∞ λ , τ F ) is a Tychonoff topological inverse semigroup.Proof. Let α and β be arbitrary elements of the semigroup I ∞ λ . We put γ = α · β and let F = { n , . . . , n i } be a finite subset of dom γ . We denote m = ( n ) α, . . . , m i = ( n i ) α and k = ( n ) γ, . . . ,k i = ( n i ) γ . Then we get that ( m ) β = k , . . . , ( m i ) β = k i . Hence we have that U α ( { n , . . . , n i } ) · U β ( { m , . . . , m i } ) ⊆ U γ ( { n , . . . , n i } )and (cid:0) U γ ( { n , . . . , n i } ) (cid:1) − ⊆ U γ − ( { k , . . . , k i } ) . Therefore the semigroup operation and the inversion are continuous in ( I (cid:31) ր∞ ( N ) , τ F ).We observe that the group of units H ( I ) of the semigroup I ∞ λ with the induced topology τ F ( H ( I ))from ( I ∞ λ , τ F ) is a topological group (see [10, pp. 313–314, Example] or [18]) and the definition ofthe topology τ F implies that every H -class of the semigroup I ∞ λ is an open-and-closed subset of thetopological space ( I ∞ λ , τ F ). Therefore Theorem 2.20 [5] implies that the topological space ( I ∞ λ , τ F )is homeomorphic to a countable topological sum of topological copies of (cid:0) H ( I ) , τ F ( H ( I )) (cid:1) . Since every T -topological group is a Tychonoff topological space (see [22, Theorem 3.10] or [8, Theorem 8.4])we conclude that the topological space ( I ∞ λ , τ F ) is Tychonoff too. This completes the proof of theproposition. (cid:3) Remark 5.12.
We observe that the topology τ F on I ∞ λ induces the discrete topology on the band E ( I ∞ λ ). Example 5.13.
We define a topology τ WF on the semigroup I ∞ λ as follows. For every α ∈ I ∞ λ wedefine a family B WF ( α ) = { U α ( F ) | F is a finite subset of dom α } , where U α ( F ) = { β ∈ I ∞ λ | dom β ⊆ dom α and ( x ) β = ( x ) α for all x ∈ F } . Since conditions (BP1)–(BP3) [7] hold for the family { B WF ( α ) } α ∈ I ∞ λ we conclude that the family { B WF ( α ) } α ∈ I ∞ λ is the base of the topology τ WF on the semigroup I ∞ λ . Proposition 5.14. ( I ∞ λ , τ WF ) is a Hausdorff topological inverse semigroup.Proof. Let α and β be arbitrary elements of the semigroup I ∞ λ . We put γ = α · β and let F = { n , . . . , n i } be a finite subset of dom γ . We denote m = ( n ) α, . . . , m i = ( n i ) α and k = ( n ) γ, . . . , k i = ( n i ) γ . Then we get that ( m ) β = k , . . . , ( m i ) β = k i . Hence we have that U α ( { n , . . . , n i } ) · U β ( { m , . . . , m i } ) ⊆ U γ ( { n , . . . , n i } )and (cid:0) U γ ( { n , . . . , n i } ) (cid:1) − ⊆ U γ − ( { k , . . . , k i } ) . Therefore the semigroup operation and the inversion are continuous in ( I ∞ λ , τ WF ).Later we shall show that the topology τ WF is Hausdorff. Let α and β be arbitrary distinct points ofthe space ( I ∞ λ , τ WF ). Then only one of the following conditions holds: N MONOIDS OF INJECTIVE PARTIAL SELFMAPS ALMOST EVERYWHERE THE IDENTITY 15 ( i ) dom α = dom β ;( ii ) dom α = dom β .In case dom α = dom β we have that there exists x ∈ dom α such that ( x ) α = ( x ) β . The definitionof the topology τ WF implies that U α ( { x } ) ∩ U β ( { x } ) = ∅ .If dom α = dom β , then only one of the following conditions holds:( a ) dom α $ dom β ;( b ) dom β $ dom α ;( c ) dom α \ dom β = ∅ and dom β \ dom α = ∅ .Suppose that case ( a ) holds. Let be x ∈ dom β \ dom α and y ∈ dom α . The definition of thetopology τ WF implies that U α ( { y } ) ∩ U β ( { x } ) = ∅ .Case ( b ) is similar to ( a ).Suppose that case ( c ) holds. Let be x ∈ dom β \ dom α and y ∈ dom α \ dom β . The definition ofthe topology τ WF implies that U α ( { y } ) ∩ U β ( { x } ) = ∅ .This completes the proof of the proposition. (cid:3) Remark 5.15.
We observe that the topology τ WF on I ∞ λ induces a non-discrete topology (and hence anon-hereditary Baire topology) on the band E ( I ∞ λ ). Moreover, H -classes in ( I ∞ λ , τ WF ) and ( I ∞ λ , τ F )are homeomorphic subspaces. Acknowledgements
We acknowledge Taras Banakh for his comments and suggestions. The authors are grateful to thereferee for several comments and suggestions which have considerably improved the original version ofthe manuscript.
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E-mail address : chuchman [email protected] Department of Mechanics and Mathematics, Ivan Franko Lviv National University, Universytetska1, Lviv, 79000, Ukraine
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