On monoids of injective partial selfmaps almost everywhere the identity

In this paper we study the semigroup $\mathscr{I}^{\infty}_\lambda$ of injective partial selfmaps almost everywhere the identity of a set of infinite cardinality $\lambda$. We describe the Green relations on $\mathscr{I}^{\infty}_\lambda$, all (two-sided) ideals and all congruences of the semigroup $\mathscr{I}^{\infty}_\lambda$. We prove that every Hausdorff hereditary Baire topology $\tau$ on $\mathscr{I}^{\infty}_\omega$ such that $(\mathscr{I}^{\infty}_\omega,\tau)$ is a semitopological semigroup is discrete and describe the closure of the discrete semigroup $\mathscr{I}^{\infty}_\lambda$ in a topological semigroup. Also we show that for an infinite cardinal $\lambda$ the discrete semigroup $\mathscr{I}^{\infty}_\lambda$ does not embed into a compact topological semigroup and construct two non-discrete Hausdorff topologies turning $\mathscr{I}^{\infty}_\lambda$ into a topological inverse semigroup.


Introduction and preliminaries
In this paper all spaces are assumed to be Hausdorff. Furthermore we shall follow the terminology of [3,5,7,9,23]. By ω we shall denote the first infinite cardinal and by |A| the cardinality of the set A. If Y is a subspace of a topological space X and A ⊆ Y , then by cl Y (A) and Int Y (A) we shall denote 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 1 (see [5]). An algebraic semigroup S is called inverse if for any element x ∈ S there exists a unique element x −1 ∈ S (called the inverse of x) such that xx −1 x = x and x −1 xx −1 = x −1 . If S is an inverse semigroup, then the function inv : S → S which assigns to every element x of S its inverse element x −1 is called inversion.
If S is an inverse semigroup, then by E(S) we shall denote the band (i.e., the subsemigroup of idempotents) 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 the existence 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 {0, −1, −2, −3, . . .} with the usual order . Let E be a semilattice 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 all finite 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]): 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 the semigroup of injective partial selfmaps almost everywhere the identity of λ. We shall denote every element α of the semigroup I ∞ λ by x 1 · · · x n y 1 · · · y n A 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 identity map; (ii) {x 1 , . . . , x n } and {y 1 , . . . , 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) transformations of topological spaces. Beȋda [2], Orlov [19,20], and Subbiah [24] have considered semigroup and inverse semigroup topologies on semigroups of partial homeomorphisms of some classes of topological spaces.
Gutik and Pavlyk [12] considered the special case of the semigroup I n λ : an infinite topological semigroup of λ × λ-matrix units B λ . They showed that an infinite topological semigroup of λ × λmatrix 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 compactification of B λ , minimal semigroup and minimal semigroup inverse topologies on B λ .
Gutik, Lawson and Repovš [11] introduced the notion of a semigroup with a tight ideal series and investigated their closures in semitopological semigroups, in particular, in inverse semigroups with continuous inversion. As a corollary they showed that the symmetric inverse semigroup of finite transformations I n λ of infinite cardinal λ is algebraically closed in the class of (semi)topological inverse semigroups 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 n λ is algebraically h-closed in the class of topological inverse semigroups. They also proved that a topological semigroup S with countably compact square S ×S does not contain the semigroup I n λ for infinite cardinals λ and showed that the Bohr compactification of an infinite topological semigroup I n λ is the trivial semigroup. In [15] Gutik and Reiter showed that that the symmetric inverse semigroup of finite transformations I n λ of infinite cardinal λ is algebraically closed in the class of semitopological inverse semigroups with continuous inversion. Also there they described all congruences on the semigroup I n λ and all compact and countably compact topologies τ on I n λ such that (I n λ , τ ) is a semitopological semigroup. Gutik, Pavlyk and Reiter [13] showed that a topological semigroup of finite partial bijections I n λ of an infinite cardinal with a compact subsemigroup of idempotents is absolutely H-closed. They proved that no Hausdorff countably compact topological semigroup and no Tychonoff topological semigroup with pseudocompact square contain I n λ as a subsemigroup. They proved that every continuous homomorphism from a topological semigroup I n λ into a Hausdorff countably compact topological semigroup or Tychonoff topological semigroup with pseudocompact square is annihilating. They also gave sufficient conditions for a topological semigroup I 1 λ to be non-H-closed and showed that the topological inverse semigroup I 1 λ is absolutely H-closed if and only if the band E(I 1 λ ) is compact [13]. In [16] Gutik and Repovš studied the semigroup I ր ∞ (N) of partial cofinite monotone bijective transformations of the set of positive integers N. They showed that the semigroup I ր ∞ (N) has algebraic properties similar to the bicyclic semigroup: it is bisimple and all of its non-trivial group homomorphisms are either isomorphisms or group homomorphisms. They proved that every locally compact topology τ on I ր ∞ (N) such that (I ր ∞ (N), τ ) is a topological inverse semigroup, is discrete and described the closure of (I ր ∞ (N), τ ) in a topological semigroup. In [4] Gutik and Chuchman studied the semigroup I ր ∞ (N) of partial co-finite almost monotone bijective transformations of the set of positive integers N. They showed that the semigroup I ր ∞ (N) has algebraic properties similar to the bicyclic semigroup: it is bisimple and all of its non-trivial group homomorphisms are either isomorphisms or group homomorphisms. Also they proved that every Baire topology τ on I ր ∞ (N) such that (I ր ∞ (N), τ ) is a semitopological semigroup is discrete, described the closure of (I ր ∞ (N), τ ) in a topological semigroup and constructed non-discrete Hausdorff semigroup topologies on the semigroup I ր ∞ (N). In this paper we study the semigroup I ∞ λ of injective partial selfmaps almost everywhere the 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 Baire topology τ on I ∞ ω such that (I ∞ ω , τ ) is a semitopological semigroup is discrete and describe the closure of 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 two non-discrete Hausdorff topologies turning I ∞ λ into a topological inverse semigroup.
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).

Statement (xiii) follows from items (viii) and (xi).
Later we shall need the following proposition: Proposition 2.3. Let λ be an arbitrary infinite cardinal. Then for every finite subset {x 1 , . . . , x n } of λ the semigroups I ∞ λ and I ∞ η are isomorphic for η = λ \ {x 1 , . . . , 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: 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 exist distinct 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 that dom α λ = dom β λ and ran α λ = ran β λ . Therefore there exists x ∈ ran α λ such that ( The obtained contradiction implies that the map h : be an arbitrary element of the semigroup I ∞ η , where A ⊆ η and x 1 , . . . , x n , y 1 , . . . , y n ∈ η. Since the map f : λ → η is bijective we conclude that 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 Therefore h is an isomorphism from the semigroup I ∞ λ onto I ∞ η . Proposition 2.4. Let λ be an arbitrary infinite cardinal. Then for every idempotent ε of the semigroup Proof. Since and ran α ⊆ ran ε}, Proposition 2.3 implies the assertion of the proposition.
Proposition 2.5. For every α, β ∈ I ∞ λ , both sets {χ ∈ I ∞ λ | α ·χ = β} and {χ ∈ I ∞ λ | χ·α = β} are finite. Consequently, every right translation and every left translation by an element of the semigroup Then S ⊆ T and the restriction of any partial map χ ∈ T to dom(α −1 · α) coincides with the partial map α −1 · β. Since every partial map from the semigroup I ∞ λ is an injective partial selfmap almost everywhere the identity we have that there exist maximal subsets A α −1 α and A α −1 β in λ such that the sets λ \ A α −1 α and λ \ A α −1 β are finite and the restrictions ( A → A are identity maps and the set λ \ A is finite. This implies that the set T is finite and hence the set S is finite too. For an arbitrary non-zero cardinal λ we denote by S ∞ (λ) the group of all bijective transformations of 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 natural homomorphisms 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: Let R is an arbitrary congruence on a semilattice E. Let a and b be elements of the semilattice E such that aRb. Then (i) aR(ab); and (ii) if a b then aRc for all c ∈ E such that a c b.
Let ι be an arbitrary idempotent of the semigroup I ∞ λ such that ι ∈ ↓ϕ. We put ι 0 = ε · ι. Then by previous part of the proof we have that ι 0 Rϕ and hence by Lemma 3.1 we get ιRϕ.
Proof. In the case when α is an idempotent of the semigroup I ∞ λ the statement of the theorem follows from Lemma 3.1 and Proposition 3.2.
Proof. Since α and β are non-H -equivalent elements of the inverse semigroup I ∞ λ we conclude that at least one of the following conditions holds: Suppose that the case α · α −1 = β · β −1 holds. In the other case the proof is similar. Since I ∞ λ is an inverse semigroup Lemma III.1.1 [21] implies that β −1 Rα −1 and hence (β · β −1 )R(α · α −1 ). Then we have that and hence the assumptions of the Theorem 3.3 hold. This completes the proof of the proposition.
We consider two cases: Suppose case (i) holds and |λ \ dom α| = n < |λ \ dom β|. Then α and β are not H -equivalent elements in I ∞ λ and hence by Proposition 3.5 we obtain that αRγ for all γ ∈ I ∞ λ with |λ\dom γ| n. Then Proposition 3.7 implies that µRν 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 αRγ for all γ ∈ I ∞ λ such that |λ \ dom γ| n. Then Proposition 3.7 implies that µRν 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 -equivalent element δ of the semigroup I ∞ λ such that αRδ. Otherwise by the previous part of the proof we have that R = K n (I). Since (α · α −1 )R(β · α −1 ) we conclude that without loss of generality we can assume that α is an identity element of H -class H(α) which contains α and β = α. Since α is an idempotent of the semigroup I ∞ λ we have that dom α = ran α and the restriction α| dom α : dom α → dom α is an identity map. Also we observe that the restriction of the partial map β| dom α : dom α → dom α is a permutation of the set dom α. Therefore without loss of generality we can consider β as a permutation of 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 the image (H(α))Φ R is singleton. Then by Theorem 2.20 [5] and Proposition 2.2 (viii) for every γ ∈ I ∞ λ with |λ \ dom γ| = |λ \ dom α| the image (H γ )Φ R of the H -class H γ which contains the element γ is singleton 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 odd permutation 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 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) we conclude that the image (H(α))Φ R is a two-element subgroup in I ∞ λ /R. Then by Theorem 2.20 [5] and Proposition 2.2 (viii) for every γ ∈ I ∞ λ with |λ \ dom γ| = |λ \ dom α| the image (H γ )Φ R of the H -class H γ which contains the element γ is a two-element subset in I ∞ λ /R and hence by Propositions 3.5, 3.6 and 3.7 we get that R = K n (A ∞ ). 4. On topologizations of the free semilattice (P <ω (λ), ∪) 4]). We shall say that a semigroup S has the F-property if for every a, b, c, d ∈ S 1 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 open neighbourhood U(x) of x such that U(x) ∩ ↓x = {x}. This is equivalent to statement that ↑x is an open subset in S.
A topological space X is called Baire if for each sequence A 1 , A 2 , . . . , A i , . . . of nowhere dense subsets A i is a co-dense subset of X [7]. A Tychonoff space X is calledČech complete if for every 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]. EveryČech 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 Hausdorff semitopological semigroup S is a closed subset of S. Hence every regular Hausdorff hereditary Baire semitopological 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 such that (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 point in (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 Then we have that S x = ∞ i=1 F n . Since the topological space S x is Baire we conclude that that there exists F n ∈ F such that Int Sx (F n ) = ∅. We fix an arbitrary y 0 ∈ Int Sx (F n ). We observe that the definition 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 0 is a local minimum in S x , i.e., ↑y 0 is an open subset of S. Since the semilattice S x has the F-property we conclude that the Hausdorffness of S implies that x is an isolated point in S x . Then x is a local minimum in S and hence ↑x is an open 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. Remark 4.4. We observe that the statement of Theorem 4.3 is true for a T 1 -semitopological I-Baire semilattice with the F-property.
Since everyČech complete (and hence locally compact) space is hereditary Baire, Theorem 4.3 implies the following corollary: Corollary 4.5. Let S be a semilattice with the F-property. Then everyČech complete (locally compact) topology τ on S such that (S, τ ) is a semitopological semilattice is discrete.
Proof. Let α be an arbitrary element of the the semigroup I ∞ ω . We need to show that α is an isolated point 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 nonempty 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 Hausdorffness of 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. Our assumption implies that I k is a closed subset of (I ∞ ω , τ ). Later we shall denote by τ k the topology induced 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 nonempty 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 a semitopological 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 −1 · V (γ) ⊆ U(γ). We remark that γ · γ −1 · V (γ) ⊆ {γ}. Hence by Proposition 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 of our theorem.
Remark 5.2. We observe that the statement of Theorem 5.1 holds for every topology τ on the semigroup 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. EveryČech 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 dense discrete 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: 2) 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 subspace of S. Suppose there exist a ∈ I ∞ λ and b ∈ I such that b · a = c / ∈ I. Since I ∞ λ is a dense open discrete subspace of S the continuity of the semigroup operation in S implies that there exists an open 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 0 = c and a 0 · 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. 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 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 there exists an accumulation point (a, b) ∈ S × S of the set D c (I ∞ λ ). The continuity of the semigroup operation in S implies that a · b = c. But I ∞ λ × I ∞ λ is a discrete subspace of S × S and hence by Theorem 5.4 the points a and b belong to the ideal I = S \ I ∞ λ and hence a · b ∈ S \ I ∞ λ cannot be equal to c.
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 if each 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 the square S × S is not pseudocompact.
A topological space X is called countably compact if any countable open cover of X contains a finite subcover [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 topological semigroup S with the countably compact square S × S.
Since every compact topological space is countably compact Theorem 3.24 [7] and Corollary 5.7 imply Corollary 5.8. For every infinite cardinal λ the discrete semigroup I ∞ λ does not embed into a compact topological semigroup.
We recall that the Stone-Čech 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 Hausdorff space 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 Tychonoff 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 the semigroup 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 the statement of the theorem.
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 we define 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 1 , . . . , n i } be a finite subset of dom γ. We denote m 1 = (n 1 )α, . . . , m i = (n i )α and k 1 = (n 1 )γ, . . . , k i = (n i )γ. Then we get that (m 1 )β = k 1 , . . . , (m i )β = k i . Hence we have that Therefore the semigroup operation and the inversion are continuous in (I 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 of the topology τ F implies that every H -class of the semigroup I ∞ λ is an open-and-closed subset of the topological 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 H(I), τ F (H(I)) . Since every T 0 -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 the proposition.