On a semigroup of closed connected partial homeomorphisms of the unit interval with a fixed point
In this paper we study the semigroup IC(I,[a]) (IO(I,[a])) of closed (open) connected partial homeomorphisms of the unit interval I with a fixed point a∈I. We describe left and right ideals of IC(I,[0]) and the Green's relations on IC(I,[0]). We show that the semigroup IC(I,[0]) is bisimple and...
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Інститут прикладної математики і механіки НАН України
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| Цитувати: | On a semigroup of closed connected partial homeomorphisms of the unit interval with a fixed point / I. Chuchman // Algebra and Discrete Mathematics. — 2011. — Vol. 12, № 2. — С. 38–52. — Бібліогр.: 10 назв. — англ. |
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irk-123456789-1548662019-06-17T01:31:06Z On a semigroup of closed connected partial homeomorphisms of the unit interval with a fixed point Chuchman, I. In this paper we study the semigroup IC(I,[a]) (IO(I,[a])) of closed (open) connected partial homeomorphisms of the unit interval I with a fixed point a∈I. We describe left and right ideals of IC(I,[0]) and the Green's relations on IC(I,[0]). We show that the semigroup IC(I,[0]) is bisimple and every non-trivial congruence on IC(I,[0]) is a group congruence. Also we prove that the semigroup IC(I,[0]) is isomorphic to the semigroup IO(I,[0]) and describe the structure of a semigroup II(I,[0])=IC(I,[0])⊔IO(I,[0]). As a corollary we get structures of semigroups IC(I,[a]) and IO(I,[a]) for an interior point a∈I. 2011 Article On a semigroup of closed connected partial homeomorphisms of the unit interval with a fixed point / I. Chuchman // Algebra and Discrete Mathematics. — 2011. — Vol. 12, № 2. — С. 38–52. — Бібліогр.: 10 назв. — англ. 1726-3255 2010 Mathematics Subject Classification:20M20,54H15, 20M18. http://dspace.nbuv.gov.ua/handle/123456789/154866 en Algebra and Discrete Mathematics Інститут прикладної математики і механіки НАН України |
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In this paper we study the semigroup IC(I,[a]) (IO(I,[a])) of closed (open) connected partial homeomorphisms of the unit interval I with a fixed point a∈I. We describe left and right ideals of IC(I,[0]) and the Green's relations on IC(I,[0]). We show that the semigroup IC(I,[0]) is bisimple and every non-trivial congruence on IC(I,[0]) is a group congruence. Also we prove that the semigroup IC(I,[0]) is isomorphic to the semigroup IO(I,[0]) and describe the structure of a semigroup II(I,[0])=IC(I,[0])⊔IO(I,[0]). As a corollary we get structures of semigroups IC(I,[a]) and IO(I,[a]) for an interior point a∈I. |
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| author |
Chuchman, I. |
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Chuchman, I. On a semigroup of closed connected partial homeomorphisms of the unit interval with a fixed point Algebra and Discrete Mathematics |
| author_facet |
Chuchman, I. |
| author_sort |
Chuchman, I. |
| title |
On a semigroup of closed connected partial homeomorphisms of the unit interval with a fixed point |
| title_short |
On a semigroup of closed connected partial homeomorphisms of the unit interval with a fixed point |
| title_full |
On a semigroup of closed connected partial homeomorphisms of the unit interval with a fixed point |
| title_fullStr |
On a semigroup of closed connected partial homeomorphisms of the unit interval with a fixed point |
| title_full_unstemmed |
On a semigroup of closed connected partial homeomorphisms of the unit interval with a fixed point |
| title_sort |
on a semigroup of closed connected partial homeomorphisms of the unit interval with a fixed point |
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Інститут прикладної математики і механіки НАН України |
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2011 |
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http://dspace.nbuv.gov.ua/handle/123456789/154866 |
| citation_txt |
On a semigroup of closed connected partial homeomorphisms of the unit interval with a fixed point / I. Chuchman // Algebra and Discrete Mathematics. — 2011. — Vol. 12, № 2. — С. 38–52. — Бібліогр.: 10 назв. — англ. |
| series |
Algebra and Discrete Mathematics |
| work_keys_str_mv |
AT chuchmani onasemigroupofclosedconnectedpartialhomeomorphismsoftheunitintervalwithafixedpoint |
| first_indexed |
2025-07-14T06:56:05Z |
| last_indexed |
2025-07-14T06:56:05Z |
| _version_ |
1837604448667435008 |
| fulltext |
Algebra and Discrete Mathematics RESEARCH ARTICLE
Volume 12 (2011). Number 2. pp. 38 – 52
c© Journal “Algebra and Discrete Mathematics”
On a semigroup
of closed connected partial homeomorphisms
of the unit interval with a fixed point
Ivan Chuchman
Communicated by M. Ya. Komarnytskyj
Abstract. In this paper we study the semigroup IC(I, [a])
(IO(I, [a])) of closed (open) connected partial homeomorphisms
of the unit interval I with a fixed point a ∈ I. We describe left
and right ideals of IC(I, [0]) and the Green’s relations on IC(I, [0]).
We show that the semigroup IC(I, [0]) is bisimple and every non-
trivial congruence on IC(I, [0]) is a group congruence. Also we
prove that the semigroup IC(I, [0]) is isomorphic to the semigroup
IO(I, [0]) and describe the structure of a semigroup II(I, [0]) =
IC(I, [0])⊔IO(I, [0]). As a corollary we get structures of semigroups
IC(I, [a]) and IO(I, [a]) for an interior point a ∈ I.
1. Introduction and preliminaries
Furthermore we shall follow the terminology of [2] and [6]. For a
semigroup S we denote the semigroup S with the adjoined unit by S1 (see
[2]).
A 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−1x = x
and x−1xx−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.
2010 Mathematics Subject Classification: 20M20,54H15, 20M18.
Key words and phrases: Semigroup of bijective partial transformations, sym-
metric inverse semigroup, semigroup of homeomorphisms, group congruence, bisimple
semigroup.
I . Chuchman 39
If S is a semigroup, then we shall denote the subset of idempotents
in S by E(S). If S is an inverse semigroup, then E(S) is closed under
multiplication and we shall refer to E(S) a band (or the band of S). If
the band E(S) is a non-empty subset of S, then the semigroup operation
on S determines the following partial order 6 on E(S): e 6 f if and only
if ef = fe = e. This order is called the natural partial order on E(S). A
semilattice is a commutative semigroup of idempotents. A semilattice E
is called linearly ordered or a chain if its natural order is a linear order.
Let E be a semilattice and e ∈ E. We denote ↓e = {f ∈ E | f 6 e} and
↑e = {f ∈ E | e 6 f}.
If S is a semigroup, then we shall denote by R, L , J , D and H the
Green relations on S (see [2]):
aRb if and only if aS1 = bS1;
aL b if and only if S1a = S1b;
aJ b if and only if S1aS1 = S1bS1;
D = L ◦ R = R ◦ L ;
H = L ∩ R.
A semigroup S is called simple if S does not contain proper two-sided
ideals and bisimple if S has only one D-class.
A congruence C on a semigroup S is called non-trivial if C is distinct
from universal and identity congruence on S, and group if the quotient
semigroup S/C is a group.
The bicyclic semigroup C (p, q) is the semigroup with the identity 1
generated by elements p and q subject only to the condition pq = 1. The
distinct elements of C (p, q) are exhibited in the following useful array:
1 p p2 p3 · · ·
q qp qp2 qp3 · · ·
q2 q2p q2p2 q2p3 · · ·
q3 q3p q3p2 q3p3 · · ·
...
...
...
...
. . .
The bicyclic semigroup is bisimple and every one of its congruences is
either trivial or a group congruence. Moreover, every non-annihilating
homomorphism h of the bicyclic semigroup is either an isomorphism or
the image of C (p, q) under h is a cyclic group (see [2, Corollary 1.32]).
The bicyclic semigroup plays an important role in algebraic theory of
semigroups and in the theory of topological semigroups. For example
the well-known Andersen’s result [1] states that a (0–)simple semigroup
40On a semigroup of closed connected partial homeomorphisms
is completely (0–)simple if and only if it does not contain the bicyclic
semigroup.
Let IX denote the set of all partial one-to-one transformations of an
non-empty set X together with the following semigroup operation:
x(αβ) = (xα)β if x ∈ dom(αβ) = {y ∈ domα | yα ∈ domβ} ,
for α, β ∈ IX . The semigroup IX is called the symmetric inverse semi-
group over the setX (see [2, Section 1.9]). The symmetric inverse semigroup
was introduced by Wagner [10] and it plays a major role in the theory of
semigroups.
Let I be an interval [0, 1] with the usual topology. A partial map
α : I ⇀ I is called:
• closed, if domα and ranα are closed subsets in I;
• open, if domα and ranα are open subsets in I;
• convex, if domα and ranα are convex non-singleton subsets in I;
• monotone, if x1 6 x2 implies (x1)α 6 (x2)α, for all x1, x2 ∈ domα;
• a local homeomorphism, if the restriction α|domα : domα → ranα
is a homeomorphism.
We fix an arbitrary a ∈ I. Hereafter we shall denote by:
• IC(I, [a]) the semigroup of all closed connected partial homeomor-
phisms α such that IntI(domα) 6= ∅ and (a)α = a;
• IO(I, [a]) the semigroup of all open connected partial homeomor-
phisms α such that (a)α = a;
• H(I) the group of all homeomorphisms of I;
• H
ր
(I) the group of all monotone homeomorphisms of I;
• I the identity map from I onto I.
Remark 1. We observe that for every a ∈ I the semigroups IC(I, [a]) and
IO(I, [a]) are inverse subsemigroups of the symmetric inverse semigroup
II over the set I.
In [3, 4] Gluskin studied the semigroup S of homeomorphic transfor-
mations of the unit interval. He described all ideals, homomorphisms and
automorphisms of the semigroup S and congruence-free subsemigroups of
S. This studies was continued in [7] by Shneperman. In [9] Shneperman
I . Chuchman 41
described the structure of the semigroup of homeomorphisms of a sim-
ple arc. In the paper [8] he studied a semigroup G(X) of all continuous
transformations of a closed subset X of the real line.
In our paper we study the semigroup IC(I, [a]) (IO(I, [a])) of closed
(open) connected partial homeomorphisms of the unit interval I with a
fixed point a ∈ I. We describe left and right ideals of IC(I, [0]) and the
Green’s relations on IC(I, [0]). We show that the semigroup IC(I, [0])
is bisimple and every non-trivial congruence on IC(I, [0]) is a group
congruence. Also we prove that the semigroup IC(I, [0]) is isomorphic
to the semigroup IO(I, [0]) and describe the structure of a semigroup
II(I, [0]) = IC(I, [0]) ⊔ IO(I, [0]). As a corollary we get structures of
semigroups IC(I, [a]) and IO(I, [a]) for an interior point a ∈ I.
2. On the semigroup IC(I, [0])
Proposition 1. The following conditions hold:
(i) every element of the semigroup IC(I, [0]) (IO(I, [1])) is a monotone
partial map;
(ii) the semigroups IC(I, [0]) and IC(I, [1]) are isomorphic;
(iii) max{domα} exists for every α ∈ IC(I, [0]);
(iv) sup{domα} exists for every α ∈ IO(I, [0]);
(v) (0)α = 0 and (1)α = 1 for every α ∈ H
ր
(I).
Proof. Statements (i), (iii), (iv) and (v) follow from elementary properties
of real-valued continuous functions.
(ii) A homomorphism i : IC(I, [0]) → IC(I, [1]) we define by the
following way:
(α)i = β, where domβ = {1− x | x ∈ domα},
ranβ = {1− x | x ∈ ranα}, and
(a)β = 1− (1− a)α for all a ∈ domβ.
Simple verifications show that such defined map i is an isomorphism from
the semigroup IC(I, [0]) onto the semigroup IO(I, [1]).
Proposition 2. The following statements hold:
(i) an element α of the semigroup IC(I, [0]) is an idempotent if and
only if (x)α = x for every x ∈ domα;
42On a semigroup of closed connected partial homeomorphisms
(ii) If ε, ι ∈ E(IC(I, [0])), then ε 6 ι if and only if dom ε ⊆ dom ι;
(iii) The semilattice E(IC(I, [0])) is isomorphic to the semilattice
((0, 1],min) under the mapping (ε)h = max{dom ε};
(iv) αRβ in IC(I, [0]) if and only if domα = domβ;
(v) αL β in IC(I, [0]) if and only if ranα = ranβ.
(vi) αH β in IC(I, [0]) if and only if domα = domβ and ranα = ranβ.
(vii) for every distinct idempotents ε, ι ∈ IC(I, [0]) there exists an element
α of the semigroup IC(I, [0]) such that α · α−1 = ε and α−1 · α = ι;
(viii) αDβ for all α, β ∈ IC(I, [0]), and hence the semigroup IC(I, [0]) is
bisimple;
(ix) αJ β for all α, β ∈ IC(I, [0]), and hence the semigroup IC(I, [0]) is
simple;
(x) a subset L is a left ideal of IC(I, [0]) if and only if there exists
a ∈ (0, 1] such that either L = {α ∈ IC(I, [0]) | ranα ⊆ [0, a)} or
L = {α ∈ IC(I, [0]) | ranα ⊆ [0, a]};
(xi) a subset R is a right ideal of IC(I, [0]) if and only if there exists a
a ∈ (0, 1] such that either R = {α ∈ IC(I, [0]) | domα ⊆ [0, a)} or
R = {α ∈ IC(I, [0]) | domα ⊆ [0, a]}.
Proof. Statements (i), (ii) and (iii) are trivial and they follow from the
definition of the semigroup IC(I, [0]).
(iv) Let be α, β ∈ IC(I, [0]) such that αRβ. Since αIC(I, [0]) =
βIC(I, [0]) and IC(I, [0]) is an inverse semigroup, Theorem 1.17 [2] implies
that
αIC(I, [0]) = αα−1IC(I, [0]) and βIC(I, [0]) = ββ−1IC(I, [0]),
and hence we have that αα−1 = ββ−1. Therefore we get that domα =
domβ.
Conversely, let be α, β ∈ IC(I, [0]) such that domα = domβ. Then
αα−1 = ββ−1. Since IC(I, [0]) is an inverse semigroup, Theorem 1.17 [2]
implies that
αIC(I, [0]) = αα−1IC(I, [0]) = ββ−1IC(I, [0]) = βIC(I, [0]),
and hence αIC(I, [0]) = βIC(I, [0]).
The proof of statement (v) is similar to (iv).
I . Chuchman 43
Statement (vi) follows from (iv) and (v).
(vii) We fix arbitrary distinct idempotents ε and ι in IC(I, [0]). If
dε = max{dom ε} and dι = max{dom ι}, then dε 6= 0, dι 6= 0, and ε and
ι are identity maps of intervals [0, dε] and [0, dι], respectively. We define a
partial map α : I ⇀ I as follows:
domα = [0, dε], ranα = [0, dι] and (x)α =
dι
dε
·x, for all x ∈ domα.
Then we have that α ∈ IC(I, [0]), α · α−1 = ε and α−1 · α = ι.
(viii) Statement (vii) and Lemma 1.1 from [5] imply that IC(I, [0]) is
a bisimple semigroup.
Since D ⊆ J , statement (viii) implies assertion (ix).
(x) The semigroup operation on IC(I, [0]) implies that the sets
{α ∈ IC(I, [0]) | ranα ⊆ [0, a)} and {α ∈ IC(I, [0]) | ranα ⊆ [0, a]} are
left ideals in IC(I, [0]), for every a ∈ (0, 1].
Suppose that L is an arbitrary left ideal of the semigroup IC(I, [0]).
We fix any α ∈ L . Then statements (i), (ii) and (v) imply that the left
ideal L contains all β ∈ L such that ranβ ⊆ ranα. We put
A =
⋃
α∈L
ranα
and let a = supA. If there exists α ∈ L such that sup ranα = a then
statement (v) implies that L = {α ∈ IC(I, [0]) | ranα ⊆ [0, a]}. In other
case we have that statement (v) implies that
L = {α ∈ IC(I, [0]) | ranα ⊆ [0, a)} .
The proof of statement (xi) is similar to statement (x).
Definitions of the group H
ր
(I) and the semigroup IC(I, [0]) imply the
following:
Proposition 3. The group of units of the semigroup IC(I, [0]) is isomor-
phic to (i.e., coincides with) the group H
ր
(I).
Proposition 2.20 of [2] states that every two subgroup which lie in some
D-class are isomorphic, and hence Proposition 3 implies the following:
Corollary 1. Every maximal subgroup of the semigroup IC(I, [0]) is
isomorphic to H
ր
(I).
Later we need the following two lemmas:
44On a semigroup of closed connected partial homeomorphisms
Lemma 1. Let R is an arbitrary congruence on a semilattice E and let
a and b be elements of the semilattice E such that aRb. If a 6 b then aRc
for all c ∈ E such that a 6 c 6 b.
The proof of the lemma follows from the definition of a congruence on
a semilattice.
Lemma 2. For arbitrary distinct idempotents α and β of the semigroup
IC(I, [0]) there exists a subsemigroup C in IC(I, [0]) such that α, β ∈ C
and C is isomorphic to the bicyclic semigroup C (p, q).
Proof. Without loss of generality we can assume that β 6 α in
E(IC(I, [0])). We define partial maps γ, δ : I ⇀ I as follows:
dom γ = [0, dα], ran γ = [0, dβ ] and (x)γ =
dβ
dα
·x, for all x ∈ dom γ,
and
dom δ = [0, dβ ], ran δ = [0, dα] and (x)δ =
dα
dβ
· x, for all x ∈ dom δ,
where dα = max{domα} and dβ = max{domβ}. Then we have that
α · γ = γ · α = γ, α · δ = δ · α = δ, γ · δ = α and δ · γ = β 6= α.
Hence by Lemma 1.31 from [2] we get that a subsemigroup in IC(I, [0])
which is generated by elements γ and δ is isomorphic to the bicyclic
semigroup C (p, q).
Theorem 1. Every non-trivial congruence on the semigroup IC(I, [0]) is
a group congruence.
Proof. Suppose that K is a non-trivial congruence on the semigroup
IC(I, [0]). Then there exist distinct elements α and β in IC(I, [0]) such
that αKβ. We consider the following three cases:
(i) α and β are idempotents in IC(I, [0]);
(ii) α and β are not H -equivalent in IC(I, [0]);
(iii) α and β are H -equivalent in IC(I, [0]).
Suppose case (i) holds and without loss of generality we assume that
α 6 β in E(IC(I, [0])). We define a partial map ρ : I ⇀ I as follows:
dom ρ = domβ, ran ρ = I and (x)ρ =
1
dβ
· x, for all x ∈ dom ρ,
I . Chuchman 45
where dβ = max{domβ}. Then we have that ρ−1 · β · ρ = I and hence
by Proposition 1(i) the element αβ = ρ−1 · α · ρ is an idempotent of the
semigroup IC(I, [0]). Obviously, αβ 6 I in E(IC(I, [0])), αβ 6= I and αβKI.
Then by Lemma 2 there exist γ, δ ∈ IC(I, [0]) such that
I·γ = γ ·I = γ, I·δ = δ·I = δ, γ ·δ = I and δ·γ = αβ 6= I,
and a subsemigroup C 〈γ, δ〉 in IC(I, [0]) which is generated by elements
γ and δ is isomorphic to the bicyclic semigroup C (p, q). Since by Corol-
lary 1.32 from [2] every non-trivial congruence on the bicyclic semigroup
C (p, q) is a group congruence on C (p, q) we get that all idempotents of
the semigroup C 〈γ, δ〉 are K-equivalent. Also by Lemma 1.31 from [2] we
get that every idempotent of the semigroup C 〈γ, δ〉 has a form
δn · γn = (δ · . . . · δ
︸ ︷︷ ︸
n−times
) · (γ · . . . · γ
︸ ︷︷ ︸
n−times
), where n = 0, 1, 2, 3, . . . ,
and hence we get that dom
(
δn · γn
)
= [0, dn], where d = max{domαβ}.
This implies that for every idempotent ε ∈ IC(I, [0]) there exists a positive
integer n such that δn ·γn 6 ε, and hence by Lemma 1 we get that all idem-
potents of the semigroup IC(I, [0]) are K-equivalent. Then Lemma 7.34
and Theorem 7.36 from [2] imply that the quotient semigroup IC(I, [0])/K
is a group.
Suppose case (ii) holds: α and β are not H -equivalent in IC(I, [0]).
Since IC(I, [0]) is an inverse semigroup we get that either αα−1 6= ββ−1
or α−1α 6= β−1β. Suppose inequality αα−1 6= ββ−1 holds. Since αKβ
and IC(I, [0]) is an inverse semigroup, Lemma III.1.1 from [6] implies
that
(
αα−1
)
K
(
ββ−1
)
, and hence by case (i) we get that K is a group
congruence on the semigroup IC(I, [0]). In the case α−1α 6= β−1β the
proof is similar.
Suppose case (iii) holds: α and β are H -equivalent in IC(I, [0]). Then
Theorem 2.3 of [2] implies that without loss of generality we can assume
that α and β are elements of the group of units H(I) of the semigroup
IC(I, [0]). Therefore we get that I = α · α−1 and γ = β · α−1 ∈ H(I) are
H -equivalent distinct elements in IC(I, [0]). Since I 6= γ we get that there
exists xγ ∈ I such that (xγ)γ 6= xγ . We suppose (xγ)γ > xγ . We define a
partial map δ : I ⇀ I as follows:
dom δ = [0, (xγ)γ], ran δ = [0, xγ ] and (x)ρ =
xγ
(xγ)γ
· x,
for all x ∈ dom δ. Then we have that I · δ = δ and hence we get that
(γ · δ)Kδ. Since dom(γ · δ) = dom γ 6= dom δ, Proposition 1(vi) implies
46On a semigroup of closed connected partial homeomorphisms
that the elements γ · δ and δ are not H -equivalent. Therefore case (ii)
holds, and hence K is a group congruence on the semigroup IC(I, [0]).
In the case (xγ)γ < xγ the proof that K is a group congruence on the
semigroup IC(I, [0]) is similar. This completes the proof of our theorem.
Proposition 4. The semigroups IC(I, [0]) and IO(I, [0]) are isomorphic.
Proof. We define a map i : IC(I, [0]) → IO(I, [0]) by the following way:
for arbitrary α ∈ IC(I, [0]) we put (α)i is the restriction of α on the set
[0, aα] \ {aα}, where aα = max{domα}, with dom((α)i) = domα \ {aα}
and ran((α)i) = ranα \ {(aα)α}. Simple verifications show that such
defined map i : IC(I, [0]) → IO(I, [0]) is an isomorphism.
3. On the semigroup II(I, [0])
We put II(I, [0]) = IC(I, [0]) ⊔ IO(I, [0]).
Later we shall denote elements of the semigroup IC(I, [0]) by α and
put
◦
α = (α)i ∈ IO(I, [0]), where i : IC(I, [0]) → IO(I, [0]) is the isomor-
phism which is defined in the proof of Proposition 4. Since the semigroups
IC(I, [0]) and IO(I, [0]) are inverse subsemigroups of the symmetric in-
verse semigroup II over the set I and by Proposition 1 all elements of the
semigroups IC(I, [0]) and IO(I, [0]) are monotone partial maps, the semi-
group operation in II implies that for α ∈ IC(I, [0]) and
◦
β ∈ IO(I, [0])
we have that
α ·
◦
β =
{
γ, if ranα ⊂ dom
◦
β;
◦
γ, if dom
◦
β ⊂ ranα
and
◦
β ·α =
{
◦
δ, if ran
◦
β ⊂ domα;
δ, if domα ⊂ ran
◦
β,
where γ = α · β ∈ IC(I, [0]) (i.e.,
◦
γ =
◦
α ·
◦
β ∈ IO(I, [0])) and δ = β · α ∈
IC(I, [0]) (i.e.,
◦
δ =
◦
β ·
◦
α ∈ IO(I, [0])). Hence we get the following:
Proposition 5. II(I, [0]) is an inverse semigroup.
Given two partially ordered sets (A,6A) and (B,6B), the lexicograph-
ical order 6lex on the Cartesian product A×B is defined as follows:
(a, b) 6lex (a′, b′) if and only if a <A a′ or (a = a′ and b 6B b′).
In this case we shall say that the partially ordered set (A × B,6lex) is
the lexicographic product of partially ordered sets (A,6A) and (B,6B)
and it is denoted by A×lex B. We observe that a lexicographic order of
two linearly ordered sets is a linearly ordered set.
I . Chuchman 47
Hereafter for every α ∈ IC(I, [0]) and
◦
β ∈ IO(I, [0]) we denote dα =
max{domα}, rα = max{ranα}, dβ = sup{dom
◦
β} and rβ = sup{ran
◦
β}.
Obviously we have that dα = sup{dom
◦
α} and rα = sup{ran
◦
α} for any
◦
α ∈ IO(I, [0]).
Proposition 6. The following conditions hold:
(i) E(II(I, [0])) = E(IC(I, [0])) ∪ E(IO(I, [0])).
(ii) If α,
◦
α, β,
◦
β ∈ E(II(I, [0])), then
(a)
◦
α 6 α;
(b) α 6 β if and only if dα 6 dβ (rα 6 rβ);
(c)
◦
α 6
◦
β if and only if dα 6 dβ (rα 6 rβ);
(d) α 6
◦
β if and only if dα < dβ (rα < rβ); and
(e)
◦
α 6 β if and only if dα 6 dβ (rα 6 rβ).
(iii) The semilattice E(II(I, [0])) is isomorphic to the lexicographic
product (0; 1] ×lex {0; 1} of the semilattices ((0; 1],min) and
({0; 1},min) under the mapping (α)i = (dα; 1) and (
◦
α)i = (dα; 0),
and hence E(II(I, [0])) is a linearly ordered semilattice.
(iv) The elements α and β of the semigroup II(I, [0]) are R-equivalent
in II(I, [0]) provides either α, β ∈ IC(I, [0]) or α, β ∈ IO(I, [0])
and moreover, we have that
(a) αRβ in II(I, [0]) if and only if dα = dβ; and
(b)
◦
αR
◦
β in II(I, [0]) if and only if dα = dβ.
(v) The elements α and β of the semigroup II(I, [0]) are L -equivalent
in II(I, [0]) provides either α, β ∈ IC(I, [0]) or α, β ∈ IO(I, [0])
and moreover, we have that
(a) αL β in II(I, [0]) if and only if rα = rβ; and
(b)
◦
αL
◦
β in II(I, [0]) if and only if rα = rβ.
(vi) The elements α and β of the semigroup II(I, [0]) are H -equivalent
in II(I, [0]) provides either α, β ∈ IC(I, [0]) or α, β ∈ IO(I, [0])
and moreover, we have that
(a) αH β in II(I, [0]) if and only if dα = dβ and rα = rβ; and
(b)
◦
αH
◦
β in II(I, [0]) if and only if dα = dβ and rα = rβ.
48On a semigroup of closed connected partial homeomorphisms
(vii) II(I, [0]) is a simple semigroup.
(viii) The semigroup II(I, [0]) has only two distinct D-classes: that are
inverse subsemigroups IC(I, [0]) and IO(I, [0]).
Proof. Statements (i), (ii) and (iii) follow from the definition of the
semigroup II(I, [0]) and Proposition 5.
Proofs of statements (iv), (v) and (vi) follow from Proposition 5
and Theorem 1.17 [2] and are similar to statements (iv), (v) and (vi) of
Proposition 2.
(vii) We shall show that II(I, [0]) · α · II(I, [0]) = II(I, [0]) for every
α ∈ II(I, [0]). We fix arbitrary α, β ∈ II(I, [0]) and show that there exist
γ, δ ∈ II(I, [0]) such that γ · α · δ = β.
We consider the following four cases:
(1) α = α ∈ IC(I, [0]) and β = β ∈ IC(I, [0]);
(2) α = α ∈ IC(I, [0]) and β =
◦
β ∈ IO(I, [0]);
(3) α =
◦
α ∈ IO(I, [0]) and β = β ∈ IC(I, [0]);
(4) α =
◦
α ∈ IO(I, [0]) and β =
◦
β ∈ IO(I, [0]).
By Λb
a we denote a linear partial map from I into I with domΛb
a =
[0; a] and ranΛb
a = [0; b], and defined by the formula: (x)Λb
a =
b
a
· x, for
x ∈ domΛb
a.
We put:
γ = Λdα
dβ
and δ = α−1 · Λ
dβ
dα
· β in case (1);
γ = Λdα
dβ
and δ = α−1 · Λ
dβ
dα
· β in case (2);
γ = Λa
dβ
and δ = α−1 · Λ
dβ
a · β, where 0 < a < dα, in case (3);
γ = Λdα
dβ
and δ = α−1 · Λ
dβ
dα
· β in case (4).
Elementary verifications show that γ · α · δ = β, and this completes the
proof of assertion (vii).
Statement (viii) follows from statements (iv) and (v).
On the semigroup II(I, [0]) we determine a relation ∼id by the follow-
ing way. Let i : IC(I, [0]) → IO(I, [0]) be a map which is defined in the
proof of Proposition 4. We put
α ∼id β if and only if α = β or (α)id = β or (β)id = α,
I . Chuchman 49
for α, β ∈ II(I, [0]). Simple verifications show that ∼id is an equivalence
relation on the semigroup II(I, [0]).
The following proposition immediately follows from Proposition 1(i)
and the definition of the relation ∼id on the semigroup II(I, [0]):
Proposition 7. Let α and β are elements of the semigroup II(I, [0]).
Then α ∼id β in II(I, [0]) if and only if the following conditions hold:
(i) dα = dβ;
(ii) rα = rβ;
(iii) (x)α = (x)β for every x ∈ [0, dα);
(iv) (y)α = (y)β for every y ∈ [0, dβ).
Proposition 8. The relation ∼id is a congruence on the semigroup
II(I, [0]). Moreover, the quotient semigroup II(I, [0])/∼id is isomorphic
to the semigroup IC(I, [0]).
Proof. We fix arbitrary α,
◦
α, β,
◦
γ ∈ II(I, [0]). It is complete to show that
the following conditions hold:
(i)
(
α · β
)
∼id
(
◦
α · β
)
;
(ii)
(
β · α
)
∼id
(
β ·
◦
α
)
;
(iii)
(
α ·
◦
γ
)
∼id
(
◦
α ·
◦
γ
)
;
(iv)
(
◦
γ · α
)
∼id
(
◦
γ ·
◦
α
)
.
Suppose case (i) holds. If dβ 6 rα, then Proposition 1(i) implies that
(x)
(
α · β
)
= (x)
(
◦
α · β
)
for all x ∈
[
0, (dβ)(α)
−1
)
, and hence by Proposi-
tion 7 we get that
(
α · β
)
∼id
(
◦
α · β
)
. If dβ > rα, then Proposition 1(i)
implies that (x)
(
α · β
)
= (x)
(
◦
α · β
)
for all x ∈ [0, dα), and hence by
Proposition 7 we get that
(
α · β
)
∼id
(
◦
α · β
)
.
In cases (ii), (iii) and (iv) the proofs are similar. Hence ∼id is a
congruence on the semigroup II(I, [0]).
Let Φ∼id
: II(I, [0]) → IC(I, [0]) a natural homomorphism which is
generated by the congruence ∼id. Since the restriction Φ∼id
|IC(I,[0]) :
IC(I, [0]) → IC(I, [0]) of the natural homomorphism Φ∼id
: II(I, [0]) →
IC(I, [0]) is an identity map we conclude that the semigroup
(II(I, [0]))Φ∼id
is isomorphic to the semigroup IC(I, [0]).
Theorem 2. Let K be a non-trivial congruence on the semigroup
II(I, [0]). Then the quotient semigroup II(I, [0])/K is either a group
or II(I, [0])/K is isomorphic to the semigroup IC(I, [0]).
50On a semigroup of closed connected partial homeomorphisms
Proof. Since the subsemigroup of idempotents of the semigroup II(I, [0])
is linearly ordered we have that similar arguments as in the proof of
Theorem 1 imply that there exist distinct idempotents ε and ι in II(I, [0])
such that εKι and ε 6 ι. If the set (ε, ι) = {υ ∈ E(II(I, [0])) | ε < υ < ι}
is non-empty, then Lemma 1 and Theorem 1 imply that the quotient
semigroup II(I, [0])/R is inverse and it contains only one idempotent,
and hence by Lemma II.1.10 from [6] we get that II(I, [0])/R is a group.
Otherwise Proposition 7(ii) implies that ε =
◦
α and ι = α for some
idempotents
◦
α ∈ IO(I, [0]) and α ∈ IC(I, [0]).
Since by Proposition 2(ix) the semigroup IC(I, [0]) is simple we get that
for every β ∈ IC(I, [0]) there exist γ, δ ∈ IC(I, [0]) such that β = γ · α · δ.
Since IC(I, [0]) is an inverse semigroup and all elements of IC(I, [0]) are
monotone partial maps of the unit interval I we conclude that that
β = β · β
−1
· γ · α · α−1 · α · α−1 · α · δ · β
−1
· β,
and hence for elements
γβ = β · β
−1
· γ · α · α−1 and δβ = α−1 · α · δ · β
−1
· β,
of the semigroup IC(I, [0]) the following conditions hold:
β = γβ · α · δβ , domβ = dom γβ , ran γβ = domα, ranα = dom δβ
and ranβ = ran δβ .
Analogously, since all elements of the semigroups IC(I, [0]) and IO(I, [0])
are monotone partial maps of I we get that
◦
β =
◦
γβ ·
◦
α ·
◦
δβ and hence
βR
◦
β. This implies that the congruence R on the semigroup II(I, [0])
coincides with the congruence ∼id on II(I, [0]). Then Proposition 8 implies
that the quotient semigroup II(I, [0])/K is isomorphic to the semigroup
IC(I, [0]).
By S2 we denote the cyclic group of order 2.
Theorem 3. For arbitrary a, b ∈ (0, 1) the semigroups IC(I, [a]) and
IC(I, [b]) are isomorphic. Moreover, for every a ∈ (0, 1) the semigroup
IC(I, [a]) is isomorphic to the direct product
S2 × IC(I, [0])× IC(I, [0]).
Proof. We fix an arbitrary a ∈ (0, 1). Obviously, the semigroup IC(I, [a])
is isomorphic to the direct product S2 × IC
ր
(I, [a]), where IC
ր
(I, [a]) is
a subsemigroup of IC(I, [a]) which consists of monotone partial maps of
the unit interval I.
I . Chuchman 51
By IC
ր
(I ⊔ I, [0]) we denote the semigroup of all monotone convex
closed partial local homeomorphisms α of the interval [−1, 1] such that
(0)α = 0 and 0 ∈ Int[−1,1](domα). We define a map i : IC
ր
(I, [a]) →
IC
ր
(I ⊔ I, [0]) by the following way. For an arbitrary α ∈ IC
ր
(I, [a]) we
determine a partial map β = (α)i ∈ IC
ր
(I ⊔ I, [0]) as follows:
(i) domβ =
[
dm(α)− a
a
,
dM (α)− a
1− a
]
, where dm(α) = min{domα}
and dM (α) = max{domα};
(ii) ranβ =
[
rm(α)− a
a
,
rM (α)− a
1− a
]
, where rm(α) = min{ranα} and
rM (α) = max{ranα}; and
(iii) (x)β =
{
(ax+ a)α, if x 6 0
((1− a)x+ a)α, if x > 0
, for all x ∈ domβ.
Simple verifications show that such defined map i : IC
ր
(I, [a]) → IC
ր
(I ⊔
I, [0]) is an isomorphism. This completes the first part of the proof of the
theorem.
Next we define a map j : IC
ր
(I ⊔ I, [0]) → IC(I, [0]) × IC(I, [0]) by
the following way. For an arbitrary α ∈ IC
ր
(I ⊔ I, [0]) we determine a
pair of partial maps (β, γ) = (α)i ∈ IC(I, [0])× IC(I, [0]) as follows:
(i) domβ = domα ∩ [0, 1] and ranβ = ranα ∩ [0, 1];
(ii) dom γ = {−x | x ∈ domα ∩ [0, 1]} and ran γ = {−x | x ∈ ranα ∩
[0, 1]};
(iii) (x)β = (x)α for x ∈ domβ; and
(iv) (x)γ = −(x)α for x ∈ dom γ.
Simple verifications show that such defined map j : IC
ր
(I ⊔ I, [0]) →
IC(I, [0])× IC(I, [0]) is an isomorphism. This completes the proof of the
theorem.
Theorem 3 implies the following:
Corollary 2. For arbitrary a, b ∈ (0, 1) the semigroups II(I, [a]) and
II(I, [b]) are isomorphic. Moreover, for every a ∈ (0, 1) the semigroup
II(I, [a]) is isomorphic to the direct product
S2 × II(I, [0])× II(I, [0]).
52On a semigroup of closed connected partial homeomorphisms
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Contact information
Ivan Chuchman Department of Mechanics and Mathematics, Ivan
Franko Lviv National University, Universytetska
1, Lviv, 79000, Ukraine
E-Mail: chuchman i@mail.ru
Received by the editors: 22.09.2011
and in final form 22.09.2011.
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