THE MONOID OF ORDER ISOMORPHISMS OF PRINCIPAL FILTERS OF A
POWER OF THE POSITIVE INTEGERS

arXiv:1802.03598v1 [] 10 Feb 2018

OLEG GUTIK AND TARAS MOKRYTSKYI
Abstract. Let n be any positive integer and IPF (Nn ) be the semigroup of all order isomorphisms
between principal filters of the n-th power of the set of positive integers N with the product order. We
study algebraic properties of the semigroup IPF (Nn ). In particular, we show that IPF (Nn ) bisimple,
E-unitary, F -inverse semigroup, describe Green’s relations on IPF (Nn ) and its maximal subgroups.
We prove that the semigroup IPF (Nn ) is isomorphic to the semidirect product of the direct n-the
power of the bicyclic monoid C n (p, q) by the group of permutation Sn , every non-identity congruence
C on the semigroup IPF (Nn ) is group and describe the least group congruence on IPF (Nn ). Also,
we show that every Hausdorff shift-continuous topology on IPF (Nn ) is discrete and discuss embedding
the semigroup IPF (Nn ) into compact-like topological semigroups.

1. Introduction and preliminaries
We shall follow the terminology of [9, 11, 13, 33, 34].
In this paper we shall denote the first infinite ordinal by ω, the set of integers by Z, the set of positive
integers by N, the set of non-negative integers by N0 , the additive group of integers by Z(+) and the
symmetric group of degree n by Sn , i.e., Sn is the group of all permutations of an n-element set. All
topological spaces, considered in this paper, are supposed to be are Hausdorff.
Let (X, 6) be a partially ordered set (a poset). For an arbitrary x ∈ X we denote
↑x = {y ∈ X : x 6 y}

and

↓x = {y ∈ X : y 6 x}

and the sets ↑x and ↓x are called the principal filter and the principal ideal, respectively, generated by the
element x ∈ X. A map α : (X, 6) → (Y, 0) from poset (X, 6) into a poset (Y, 0) is called monotone or
order preserving if x 6 y in (X, 6) implies that xα 0 yα in (Y, 0). A monotone map α : (X, 6) → (Y, 0)
is said to be order isomorphism if it is bijective and its converse α−1 : (Y, 0) → (X, 6) is monotone.
An semigroup S is called inverse if for any element x ∈ S there exists a unique x−1 ∈ S such that
xx−1 x = x and x−1 xx−1 = x−1 . The element x−1 is called the inverse of x ∈ S. 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 the inversion.
A congruence C on a semigroup S is called non-trivial if C is distinct from universal and identity
congruences on S, and a group congruence if the quotient semigroup S/C is a group.
If S is a semigroup, then we shall denote the subset of all idempotents in S by E(S). If S is an
inverse semigroup, then E(S) is closed under multiplication and we shall refer to E(S) as a band (or
the band of S). Then the semigroup operation on S determines the following partial order 4 on E(S):
e 4 f if and only if ef = f e = 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. By (P(λ), ∩) we shall denote the semilattice of all subsets of a set of
cardinality λ > ω with the semilattice operation “intersection”.
Date: February 13, 2018.
2010 Mathematics Subject Classification. 20M18, 20M20, 22A15, 54D40, 54D45, 54H10.
Key words and phrases. Semigroup, partial map, inverse semigroup, least group congruence, permutation semigroup,
bicyclic monoid, semidirect product, semitopological semigroup, topological semigroup,
1

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OLEG GUTIK AND TARAS MOKRYTSKYI

If S is a semigroup, then we shall denote the Green relations on S by R, L , J , D and H (see
[11]). A semigroup S is called simple if S does not contain proper two-sided ideals and bisimple if S
has only one D-class.
Hereafter we shall assume that λ is an infinite cardinal. If α : λ ⇀ λ is a partial map, then we shall
denote the domain and the range of α by dom α and ran α, respectively.
Let Iλ denote the set of all partial one-to-one transformations of an infinite set X of cardinality λ
together with the following semigroup operation:
x(αβ) = (xα)β

if x ∈ dom(αβ) = {y ∈ dom α | yα ∈ dom β} ,

for α, β ∈ Iλ .

The semigroup Iλ is called the symmetric inverse semigroup over the set X (see [11, Section 1.9]). The
symmetric inverse semigroup was introduced by Wagner [38] and it plays a major role in the theory of
semigroups.
The bicyclic semigroup (or the bicyclic monoid ) C (p, q) is the semigroup with the identity 1 generated
by elements p and q subject only to the condition pq = 1.
Remark 1.1. We observe that the bicyclic semigroup is isomorphic to the semigroup CN (α, β) which
is generated by partial transformations α and β of the set of positive integers N, defined as follows:
(n)α = n + 1 if n > 1 and (n)β = n − 1 if n > 1 (see Exercise IV.1.11(ii) in [33]).
If T is a semigroup, then we say that a subsemigroup S of T is a bicyclic subsemigroup of T if S is
isomorphic to the bicyclic semigroup C (p, q).
A semitopological (topological ) semigroup is a topological space with separately continuous (continuous) semigroup operations. An inverse topological semigroup with continuous inversion is called a
topological inverse semigroup.
A topology τ on a semigroup S is called:
• semigroup if (S, τ ) is a topological semigroup;
• shift-continuous if (S, τ ) is a semitopological semigroup.
The bicyclic semigroup is bisimple and every one of its congruences is either trivial or a group
congruence. Moreover, every homomorphism h of the bicyclic semigroup is either an isomorphism or
the image of C (p, q) under h is a cyclic group (see [11, Corollary 1.32]). The bicyclic semigroup plays
an important role in algebraic theory of semigroups and in the theory of topological semigroups. For
example a well-known Andersen’s result [1] states that a (0–)simple semigroup with an idempotent is
completely (0–)simple if and only if it does not contain an isomorphic copy of the bicyclic semigroup. The
bicyclic monoid admits only the discrete semigroup Hausdorff topology and if a topological semigroup
S contains it as a dense subsemigroup then C (p, q) is an open subset of S [12]. We observe that the
openness of C (p, q) in its closure easily follows from the non-topologizability of the bicyclic monoid,
because the discrete subspace D is open in its closure D in any T1 -space containing D. Bertman and
West in [8] extended this result for the case of Hausdorff semitopological semigroups. Stable and Γcompact topological semigroups do not contain the bicyclic monoid [2, 26]. The problem of embedding
the bicyclic monoid into compact-like topological semigroups was studied in [4, 5, 22]. Independently
to Eberhart-Selden results on topolozabilty of the bicyclic semigroup, in [36] Taimanov constructed a
commutative semigroup Aκ of cardinality κ which admits only the discrete semigroup topology. Also,
Taimanov [37] gave sufficient conditions on a commutative semigroup to have a non-discrete semigroup
topology. In the paper [16] it was showed that for the Taimanov semigroup Aκ from [36] the following
conditions hold: every T1 -topology τ on the semigroup Aκ such that (Aκ , τ ) is a topological semigroup
is discrete; for every T1 -topological semigroup which contains Aκ as a subsemigroup, Aκ is a closed
subsemigroup of S; and every homomorphic non-isomorphic image of Aκ is a zero-semigroup.
Also, in the paper [14] it is proved that the discrete topology is the unique shift-continuous Hausdorff
topology on the extended bicyclic semigroup CZ . We observe that for many (0-) bisimple semigroups of
transformations S the following statement holds: every shift-continuous Hausdorff Baire (in particular

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3

locally compact) topology on S is discrete (see [10, 21, 23, 24]). In the paper [31] Mesyan, Mitchell,
Morayne and Péresse showed that if E is a finite graph, then the only locally compact Hausdorff
semigroup topology on the graph inverse semigroup G(E) is the discrete topology. In [7] it was proved
that the conclusion of this statement also holds for graphs E consisting of one vertex and infinitely many
loops (i.e., infinitely generated polycyclic monoids). Amazing dichotomy for the bicyclic monoid with
adjoined zero C 0 = C (p, q) ⊔ {0} was proved in [15]: every Hausdorff locally compact semitopological
bicyclic monoid C 0 with adjoined zero is either compact or discrete. The above dichotomy was extended
by Bardyla in [6] to locally compact λ-polycyclic semitopological monoids and to locally compact
semitopological interassociates of the bicyclic monoid with adjoined zero [17].
We observe that some classes of inverse semigroups of partial transformations with cofinite domains
and range have algebraic properties similar to the bicyclic semigroup. In [25] it was showed that the
semigroup Iλcf of injective partial cofinite selfmaps of cardinal λ is a bisimple inverse semigroup and
that for every non-empty chain L in E(Iλcf ) there exists an inverse subsemigroup S of Iλcf such that
S is isomorphic to the bicyclic semigroup and L ⊆ E(S), and proved that every non-trivial congruence
on Iλcf is a group congruence. Also, there described the structure of the quotient semigroup Iλcf /σ,
where σ is the least group congruence on Iλcf .
ր
ր
The semigroups I∞
(N) and I∞
(Z) of injective isotone partial selfmaps with cofinite domains and
images of positive integers and integers, respectively, were studied in [23] and [24]. There it was proved
ր
ր
that the semigroups I∞
(N) and I∞
(Z) are bisimple and every non-trivial homomorphic image of
ր
ր
ր
I∞
(N) and I∞
(Z) is a group, and moreover the semigroup I∞
(N) has Z(+) as a maximal group
ր
image and I∞ (Z) has Z(+) × Z(+), respectively.
In [21] the semigroup IO∞ (Znlex ) of monotone injective partial selfmaps of the set of Ln ×lex Z having
cofinite domain and image, where Ln ×lex Z is the lexicographic product of n-elements chain and the set
of integers with the usual linear order is studied. There it was showed that the semigroup IO∞ (Znlex )
is bisimple and established its projective congruences. Also, there it was proved that IO∞ (Znlex ) is
finitely generated, every automorphism of IO∞ (Z) is inner and showed that in the case n > 2 the
semigroup IO∞ (Znlex ) has non-inner automorphisms. In [21] we proved that for every positive integer
n the quotient semigroup IO∞ (Znlex )/Cmg , where Cmg is a least group congruence on IO∞ (Znlex ), is
isomorphic to the direct power (Z(+))2n . The structure of the sublattice of congruences on IO∞ (Znlex )
which contained in the least group congruence is described in [18].
In the other side the semigroup of monotone injective partial selfmaps cofinite domains and images of
the square N2 with the product order has more difficult properties to above described inverse semigroups
[19]. In particularly, there was proved that D = J in PO∞ (N26 ). In [20] it was proved that the natural
partial order 4 on PO∞ (N26 ) coincides with the natural partial order which is induced from symmetric
inverse monoid IN×N over N × N onto the semigroup PO∞ (N26 ). The congruence σ on the semigroup
PO∞ (N26 ), which is generated by the natural order 4 on the semigroup PO∞ (N26 ) is described: ασβ if
and only if α and β are comparable in PO∞ (N26 ), 4 . Also, there was proved that quotient semigroup
PO∞ (N26 )/σ is isomorphic to the semidirect product of the free commutative monoid AMω over an
infinite countable set by the cyclic group of the order two Z2 .
For an arbitrary positive integer n by (Nn , 6) we denote the n-th power of the set of positive integers
N with the product order:
(x1 , . . . , xn ) 6 (y1 , . . . , yn )

if and only if

xi ≤ yi

for all i = 1, . . . , n.

It is obvious that the set of all order isomorphisms between principal filters of the poset (Nn , 6) with
the operation of composition of partial maps form a semigroup. This semigroup we shall denote by
IPF (Nn ). It is easy to see that the semigroup IPF (Nn ) is isomorphic to the Munn semigroup of Nn
with the dual order to 6 (see [27, Section 5.4] or [32]). Also, Remark 1.1 implies that the semigroup
IPF (Nn ) is a some generalization of the bicyclic semigroup C (p, q). Hence it is natural to ask: what
algebraic properties of the semigroup IPF (Nn ) are similar to ones of the bicyclic monoid?

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OLEG GUTIK AND TARAS MOKRYTSKYI

Later by I we shall denote the identity map of Nn and it is obvious that I is unit element of the
semigroup IPF (Nn ). Also by H(I) we shall denote the group of units of IPF (Nn ) and its clear that
α ∈ IPF (Nn ) is an element of H(I) if and only if it is an order isomorphism of the poset (Nn , 6).
In this paper we study algebraic properties of the semigroup IPF (Nn ). In particular, we show that
IPF (Nn ) bisimple, E-unitary, F -inverse semigroup, describe Green’s relations on IPF (Nn ) and its
maximal subgroups. We prove that the semigroup IPF (Nn ) is isomorphic to the semidirect product
of the n-th direct power of the bicyclic monoid C n (p, q) by the group of permutation Sn , every nonidentity congruence C on the semigroup IPF (Nn ) is group and describe the least group congruence
on IPF (Nn ). Also, we show that every Hausdorff shift-continuous topology on IPF (Nn ) is discrete
and discuss embedding the semigroup IPF (Nn ) into compact-like topological semigroups.
2. Algebraic properties of the semigroup IPF (Nn )
By P↑ (Nn ) we denote the set of all principal filters of the poset (Nn , 6). It is obvious that the
semilattice operation of (P(Nn ), ∩) is closed on P↑ (Nn ), and hence (P↑ (Nn ), ∩) is a semilattice. Also,
we observe that the semilattice (Nn , max), which is the set Nn with the point-wise operation max:
(x1 , . . . , xn ) (y1 , . . . , yn ) = (max {x1 , y1} , . . . , max {xn , yn }) ,
is isomorphic to the semilattice (P↑ (Nn ), ∩) by the mapping (x1 , . . . , xn ) 7→ ↑ (x1 , . . . , xn ).
Proposition 2.1. Let n be any positive integer. Then the following statements hold:
(i) IPF (Nn ) is an inverse semigroup;
(ii) the semilattice E(IPF (Nn )) is isomorphic to (P↑ (Nn ), ∩) by the mapping ε 7→ dom ε, and
hence it is isomorphic to the semilattice (Nn , max);
(iii) αL β in IPF (Nn ) if and only if dom α = dom β;
(iv) αRβ in IPF (Nn ) if and only if ran α = ran β;
(v) αH β in IPF (Nn ) if and only if dom α = dom β and ran α = ran β;
(vi) for any idempotents ε, ι ∈ IPF (Nn ) there exist elements α, β ∈ IPF (Nn ) such that αβ = ε
and βα = ι;
(vii) IPF (Nn ) is bisimple and hence it is simple.
Proof. It it obvious that IPF (Nn ) is an inverse subsemigroup of the symmetric inverse monoid INn
over the set Nn . Then statements (i)–(v) follow from the definitions of the semigroup IPF (Nn ) and
Green’s relations.
(vi) Fix arbitrary idempotents ε, ι ∈ IPF (Nn ). Since dom ε and dom ι are principal filters in (Nn , 6)
we put dom ε and dom ι are generated by elements (x01 , . . . , x0n ) and (y10 , . . . , yn0 ) of the poset (Nn , 6).
We define a partial map α : Nn ⇀ Nn in the following way:
dom α = dom ε,

ran α = dom ι
and

(z1 , . . . , zn ) α = z1 − x01 + y10, . . . , zn − x0n + yn0 , for any (z1 , . . . , zn ) ∈ dom α.

Then αα−1 = ε and α−1 α = ι and hence we put β = α−1 .
Statement (vii) follows from (vi) and Proposition 3.2.5(1) of [29].



The proofs of the following two lemmas are trivial.
Lemma 2.2. Let n be a positive integer ≥ 2. Then for any i = 1, . . . , n the projection on i-th coordinate
πi : Nn → Nn : (x1 , . . . , xi , . . . , xn ) 7−→ (1, . . . , xi , . . . , 1)
|{z}
|{z}
i-th

i-th

is a monotone map and moreover (1, . . . , xi , . . . , 1) 6 (x1 , . . . , xi , . . . , xn ) in (Nn , 6).
|{z}
|{z}
i-th

i-th

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5

Lemma 2.3. Let n be a positive integer ≥ 2. Then every map α : Nn → Nn which permutates the
coordinates of elements of Nn is a n order isomorphism of the poset (Nn , 6).
Lemma 2.4. Let n be a positive integer ≥ 2 and a map α : Nn → Nn be an order isomorphism of the
poset (Nn , 6) such that (x2i ) α = x2i for any i = 1, . . . , n, where x2i = (1, . . . , |{z}
2 , . . . , 1) is the element
i-th

of the poset (Nn , 6) such that only i-th coordinate of x2i is equal to 2 and all other coordinates are equal
to 1. Then α is the identity map of (Nn , 6).

Proof. We observe that the element (1, 1, . . . , 1) is the minimum in the poset (Nn , 6) and since α is an
order isomorphism of (Nn , 6) we have that (1, 1, . . . , 1)α = (1, 1, . . . , 1).
Now, by induction we show that for any positive integer k > 2 for the element xk1 = (k, 1, 1, . . . , 1)
of (Nn , 6) the set ↓xk1 is a k-element chain, then so is (xk1 )α, which implies that only first coordinate of
(xk1 )α is equal to k and all other coordinates are equal to 1. Similarly, for every positive i = 1, . . . , n by
induction we show that for any positive integer k > 2 for the element xki of (Nn , 6) with the property
that only i-th coordinate of xki is equal to k and all other coordinates are equal to 1 the set ↓xki is a
k-element chain, then so is (xki )α, which implies that only i-th coordinate of (xki )α is equal to k and all
other coordinates are equal to 1.
Hence we shown that (xpi ) α = xpi for any i = 1, . . . , n, where xpi = (1, . . . , p , . . . , 1) is the element
|{z}
i-th

of (Nn , 6) such that only i-th coordinate of xpi is equal to p and all other coordinates are equal to 1.
We observe that the converse map α−1 : Nn → Nn to the order isomorphism α of (Nn , 6) is an order
isomorphism of (Nn , 6) as well, which satisfies the assumption of lemma. Hence the above part of the
proof implies that (xpi ) α−1 = xpi for any i = 1, . . . , n, where xpi = (1, . . . , p , . . . , 1) is the element of
|{z}
i-th

(Nn , 6) such that only i-th coordinate of x2i is equal to p and all other coordinates are equal to 1.
Fix an arbitrary (x1 , . . . , xn ) ∈ Nn and suppose that
(x1 , . . . , xn )α = (y1 , . . . , yn ).

Since a composition of monotone maps is again a monotone map, Lemma 2.2 and the above part of the
proof imply that for any i = 1, . . . , n we have that
(1, . . . , xi , . . . , 1) = (1, . . . , xi , . . . , 1)πi = ((1, . . . , xi , . . . , 1)α)πi 6
|{z}
|{z}
|{z}
i-th

i-th

i-th

6 ((x1 , . . . , xn )α)πi = (y1 , . . . , yn )πi = (1, . . . , yi , . . . , 1)
|{z}
i-th

and

(1, . . . , yi , . . . , 1) = (1, . . . , yi , . . . , 1)πi = ((1, . . . , yi , . . . , 1)α−1 )πi 6
|{z}
|{z}
|{z}
i-th

i-th

i-th

−1

6 ((y1 , . . . , yn ))α πi = ((x1 , . . . , xn )α)πi = (1, . . . , xi , . . . , 1).
|{z}
i-th

This implies that (x1 , . . . , xn ) = (y1 , . . . , yn ), which completes the proof of the lemma.



Theorem 2.5. For any positive integer n the group of units H(I) of the semigroup IPF (Nn ) is
isomorphic to Sn . Moreover, every element of H(I) permutates coordinates of elements of Nn , and only
so permutations are elements of H(I).
Proof. We shall show that every order isomorphism of the poset (Nn , 6) permutates coordinates of
elements in Nn . The converse statement follows from Lemma 2.3.

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OLEG GUTIK AND TARAS MOKRYTSKYI

Next we consider the element x21 = (2, 1, 1, . . . , 1) of (Nn , 6). Since ↓x21 is a chain which consists
of two elements and α is an order isomorphism of (Nn , 6), the image (x21 )α has above properties so
x21 . Then the definition of poset (Nn , 6) the element (x21 )α satisfies the following property: only one
coordinate of (x21 )α is equal to 2 and all other coordinates are equal to 1. Such coordinate of (x21 )α we
denote by σ1 . Similar arguments show that for every element x2i of the poset (Nn , 6) with the property
that only i-th coordinate of x2i is equal to 2 and all other coordinates are equal to 1 we obtain that only
σi -th coordinate of (x2i )α is equal to 2 and all other coordinates are equal to 1, for all i = 1, . . . , n. Also,
since α is an order isomorphism of (Nn , 6) we get that σi = σj if and only if i = j, for i, j = 1, . . . , n.
Thus we defined a permutation σ : {1, . . . , n} → {1, . . . , n}. Then the compositions ασ −1 and σ −1 α
are order isomorphisms of (Nn , 6). Moreover ασ −1 and σ −1 α satisfy the assumption of Lemma 2.4,
which implies that the maps ασ −1 : Nn → Nn and σ −1 α : Nn → Nn are the identity maps of Nn . This
implies that α = σ which completes the proof the theorem.

Theorems 2.3 and 2.20 from [11] and Theorem 2.5 imply the following corollary.
Corollary 2.6. Let n be any positive integer. Then every maximal subgroup of IPF (Nn ) is isomorphic
to Sn and every H -class of IPF (Nn ) consists of n! elements.
The following proposition gives sufficient conditions when a subsemigroup of the semigroup IPF (Nn ),
which generated by an element of IPF (Nn ) and its inverse, is isomorphic to the bicyclic monoid C (p, q).
Proposition 2.7. Let n be any positive integer and α be an element of the semigroup IPF (Nn ) such
that ran α $ dom α. Then the subsemigroup hα, α−1 i of IPF (Nn ), which is generated by α and its
inverse α−1 , is isomorphic to the bicyclic monoid C (p, q).
Proof. Put ε be the identity map of dom α. The semigroup operation of IPF (Nn ) implies that the
following equalities hold:
εα = αε = α,

εα−1 = α−1 ε = α−1 ,

αα−1 = ε

and

α−1 α 6= ε

Then we apply Lemma 1.31 from [11].



Corollary 2.8. Let n be any positive integer. Then for any idempotents ε and ι of the semigroup
IPF (Nn ) such that ε 4 ι there exists a subsemigroup C of IPF (Nn ) which is isomorphic to the
bicyclic monoid C (p, q) and contains ε and ι.
Proof. Suppose that ε 6= ι. Let α be any order isomorphism from dom ι onto dom ε. Next we apply
Proposition 2.7.
If ε = ι then we choice any idempotent ν 6= ε such that ν 4 ε and apply the above part of the
proof.

Lemma 2.9. Let n be any positive integer ≥ 2 and C be a congruence on the semigroup IPF (Nn )
such that εCι for some two distinct idempotents ε, ι ∈ IPF (Nn ). Then ςCυ for all idempotents ς, υ of
IPF (Nn ).
Proof. We observe that without loss of generality we may assume that ε 4 ι where 4 is the natural
partial order on the semilattice E(IPF (Nn )). Indeed, if ε, ι ∈ E(IPF (Nn )) then εCι implies that
ε = εεCιε, and since the idempotents ε and ι are distinct in IPF (Nn ) we have that ιε 4 ε.
Now, the inequality ε 4 ι implies that dom ε ⊆ dom ι and hence (x1 , . . . , xn ) 6 (y1 , . . . , yn ), where
↑ (x1 , . . . , xn ) and ↑ (y1 , . . . , yn ) are principal filters in (Nn , 6) such that
↑ (x1 , . . . , xn ) = dom ι

and

↑ (y1 , . . . , yn ) = dom ε.

Next, we define partial maps α, β : Nn ⇀ Nn in the following way:
(a) dom α = Nn , ran α = dom ι and (z1 , . . . , zn ) α = (z1 + x1 − 1, . . . , zn + xn − 1) for (z1 , . . . , zn ) ∈
dom α;

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7

(b) dom β = dom ι, ran β = Nn and (z1 , . . . , zn ) β = (z1 − x1 + 1, . . . , zn − xn + 1) for (z1 , . . . , zn ) ∈
dom β.
Simple verifications show that αιβ = I and βα = ι, and moreover since αβ = I we have that
(αεβ) (αεβ) = αε (βα) εβ = αειεβ = αεεβ = αεβ,
which implies that αεβ is an idempotent of IPF (Nn ) such that αεβ 6= I.
Thus, it was shown that there exists a non-unit idempotent ε∗ in IPF (Nn ) such that ε∗ CI. This
implies that ε0 CI for any idempotent ε0 of IPF (Nn ) such that ε∗ 4 ε0 4 I. Then the definition of the
semigroup IPF (Nn ) and Proposition 2.1(ii) imply that there exists an element x2i = (1, . . . , |{z}
2 , . . . , 1)
i-th

of the poset (Nn , 6) such that only i-th coordinate of x2i is equal to 2 and all other coordinates are
equal to 1 and there exists an idempotent εi such that dom ε0 ⊆ dom εi = ↑x2i .
Fix an arbitrary positive integer j ∈ {1, . . . , n} \ {i}. Put σ(i,j) is the permutation of coordinates of
elements of the set Nn which permutates only j-th and i-th coordinates, i.e., this permutation is the
cycle (i, j) on the coordinates. Then the semigroup operation of IPF (Nn ) implies that σ(i,j) Iσ(i,j) = I
and σ(i,j) εi σ(i,j) = εj , where εj is the identity map of the principal filter ↑x2j of the poset (Nn , 6) such
that x2j = (1, . . . , |{z}
2 , . . . , 1), i.e., only j-th coordinate of x2j is equal to 2 and all other coordinates are
j -th

equal to 1.
The above arguments imply that εi CI for every idempotent εi ∈ IPF (Nn ) such that εi is the identity
map of the principal filter ↑x2i of the poset (Nn , 6), i = 1, . . . , n. This implies that IC (ε1 . . . εn ). The
semigroup operation of IPF (Nn ) implies that the idempotent ε1 . . . εn is the identity map of the
principal filter ↑(2, . . . , 2) of (Nn , 6). We define a partial map γ : Nn ⇀ Nn in the following way:
dom γ = Nn ,

ran γ = ↑(2, . . . , 2)

and

(z1 , . . . , zn ) γ = (z1 + 1, . . . , zn + 1) ,

for (z1 , . . . , zn ) ∈ dom γ. By Proposition 2.7 the subsemigroup hγ, γ −1 i of IPF (Nn ), which is generated
by γ and its inverse γ −1 , is isomorphic to the bicyclic monoid C (p, q). It is obvious that γγ −1 = I and
γ −1 γ = ε1 . . . εn . Since IC (ε1 . . . εn ), by Corollary 1.32 from [11] we obtain that all idempotents of
the subsemigroup hγ, γ −1 i in IPF (Nn ) are C-equivalent. Also, the definition of the bicyclic semigroup
C (p, q) and Lemma 1.31 from [11] imply that all idempotents of the subsemigroup hγ, γ −1 i of IPF (Nn )
k
are elements of the form (γ −1 ) γ k , where k is a some non-negative integer. Now, by the definition of
k
the semigroup IPF (Nn ) we have that (γ −1 ) γ k is the identity map of the principal filter ↑(k, . . . , k)
of (Nn , 6) for some non-negative integer k. Moreover, for every idempotent ζ of IPF (Nn ) which is
m
the identity map of the principal filter ↑(a1 , . . . , an ) of (Nn , 6), we have that (γ −1 ) γ m 4 ζ, where
m = max {a1 , . . . , an } ,
which implies that ICζ.



Lemma 2.10. Let n be any positive integer ≥ 2 and C be a congruence on the semigroup IPF (Nn )
such that αCβ for some non-H -equivalent elements α, β ∈ IPF (Nn ). Then εCι for all idempotents
ε, ι of IPF (Nn ).
Proof. Since α and β are not-H -equivalent in IPF (Nn ) we have that either αα−1 6= ββ −1 or
α−1 α 6= β −1 β (see [29, p. 82]). Then Proposition 4 from [29, Section 2.3] implies that αα−1 Cββ −1
and α−1 αCβ −1 β and hence the assumption of Lemma 2.9 holds.

Lemma 2.11. Let n be any positive integer ≥ 2 and C be a congruence on the semigroup IPF (Nn ) such
that αCβ for some two distinct H -equivalent elements α, β ∈ IPF (Nn ). Then εCι for all idempotents
ε, ι of IPF (Nn ).

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OLEG GUTIK AND TARAS MOKRYTSKYI

Proof. By Proposition 2.1(vii) the semigroup IPF (Nn ) is simple and then Theorem 2.3 from [11]
implies that there exist µ, ξ ∈ IPF (Nn ) such that f : Hα → HI : χ 7→ µχξ maps α to I and β to
γ 6= I, respectively, which implies that ICγ. Since γ 6= I is an element of the group of units of the
semigroup IPF (Nn ), by Theorem 2.5, γ permutates coordinates of elements of Nn , and hence there
2 , . . . , 1) is the element of the
exists a positive integer iγ such that (x2i ) α 6= x2iγ , where x2iγ = (1, . . . , |{z}
iγ -th

poset (Nn , 6) with the property that only iγ -th coordinate of x2iγ is equal to 2 and all other coordinates
are equal
to 1. Also, by Theorem 2.5 there exists a positive integer jγ ∈ {1, . . . , n} \ {iγ } such that

2
2 , . . . , 1) is the element of the poset (Nn , 6) with the property that only jγ -th
xiγ γ = x2jγ = (1, . . . , |{z}
jγ -th

coordinate of x2jγ is equal to 2 and all other coordinates are equal to 1.
Put ε is the identity map of the principal filter ↑x2iγ . Since C is a congruence on the semigroup
IPF (Nn ) and γ ∈ HI we have that
ε = εε = εIεCεγε.
Since jγ 6= iγ , the semigroup operation of IPF (Nn ) implies that dom(εγε) $ dom ε. Then by Proposition 2.1(v), εγε and ε are non-H -equivalent elements in IPF (Nn ). Next we apply Lemma 2.10. 
Theorem 2.12. Let n be any positive integer ≥ 2. Then every non-identity congruence C on the
semigroup IPF (Nn ) is group.
Proof. For every non-identity congruence C on IPF (Nn ) there exist two distinct elements α, β ∈
IPF (Nn ) such that αCβ. If αH β in IPF (Nn ) then by Lemma 2.11 all idempotents of the semigroup
IPF (Nn ) are C-equivalent, otherwise by Lemma 2.10 we get same. Thus, by Lemma II.1.10 of [33]
the quotient semigroup IPF (Nn )/C has a unique idempotent and hence it is a group.


For arbitrary elements x = (x1 , . . . , xn ) and y = (y1 , . . . , yn ) of Nn and any permutation σ : {1, . . . , n}
→ {1, . . . , n} we denote

(x)σ = x(1)σ−1 , . . . , x(n)σ−1 ;
max{x, y} = (max{x1 , y1 }, . . . , max{xn , yn }) .

Lemma 2.13. For every positive integer n and any x, y ∈ Nn the following conditions hold:
(i) (x + y)σ = (x)σ + (y)σ;
(ii) (x − y)σ = (x)σ − (y)σ;
(iii) (max{x, y})σ = max{(x)σ, (y)σ}.
Proof. In (i) we have that
(x + y)σ = (x1 + y1 , . . . , xn + yn )σ = (p1 , . . . , pn )σ
for pi = xi + yi , i = 1, . . . , n, and then
(p1 , . . . , pn )σ = (p(1)σ−1 , . . . , p(n)σ−1 ) = (x(1)σ−1 + y(1)σ−1 , . . . , x(n)σ−1 + y(n)σ−1 ) = (x)σ + (y)σ.
The proofs of (ii) and (iii) are similar.



The statement of the following lemma follows from the definition of the semigroup IPF (Nn ).
Lemma 2.14. For every α ∈ IPF (Nn ) there exist unique x, y ∈ Nn , ρα , λα ∈ IPF (Nn ) and σα ∈ Sn
such that α = ρα σα λα and
dom ρα = dom α = ↑x,
ran λα = ran α = ↑y,

ran ρα = Nn ,
n

dom λα = N ,

(z)ρα = z − x + 1

for z ∈ dom ρα ;

(z)λα = z + y − 1

for z ∈ dom λα ,

where 1 = (1, . . . , 1) is the smallest element of the poset (Nn , 6).

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9

Later in this section, for every α ∈ IPF (Nn ) by ρα , λα and σα we denote the elements ρα , λα ∈
IPF (Nn ) and σα ∈ Sn whose are determined in Lemma 2.14.
Lemma 2.15. Let α and β be elements of the semigroup IPF (Nn ) such that dom α = ↑x, ran α = ↑y,
dom β = ↑u and ran β = ↑v. Then
dom(αβ) = ↑[(max{y, u} − y)σα−1 + x];
ran(αβ) = ↑[(max{y, u} − u)σβ + v];
σαβ = σα σβ .
Proof. The definition of the domain of the composition of partial transformations (see [29, p. 4]) implies
that
dom(αβ) = [ran α ∩ dom β]α−1 = [↑y ∩ ↑u]α−1 = [↑ max{y, u}]α−1.
Since α is a monotone bijection between principal filters of the poset (Nn , 6) we get that

[↑ max{y, u}]α−1 = ↑ [max{y, u}]α−1 ,
and by Lemma 2.14

dom(αβ) = ↑ [max{y, u}]α−1 =


−1 −1
= ↑ [max{y, u}]λ−1
=
α σα ρα


= ↑ [max{y, u} − y + 1]σα−1 ρ−1
=
α

= ↑ [(max{y, u} − y)σα−1 + 1]ρ−1
=
α
= ↑[(max{y, u} − y)σα−1 + x].

Similar, the definition of the range of the composition of partial transformations (see [29, p. 4]) implies
that
ran(αβ) = [ran α ∩ dom β]β = [↑y ∩ ↑u]β = [↑ max{y, u}]β.
Since β is a monotone bijection between principal filters of the poset (Nn , 6) we get that
[↑ max{y, u}]β = ↑ ([max{y, u}]β) ,
and by Lemma 2.14
ran(αβ) = ↑ ([max{y, u}]β) =
= ↑ ([max{y, u}]ρβ σβ λβ ) =
= ↑ ([max{y, u} − u + 1]σβ λβ ) =
= ↑ ([(max{y, u} − u)σβ + 1]λβ ) =
= ↑[(max{y, u} − u)σβ + v].
We observe that definitions of elements σα and σβ imply that
dom σα = ran σα = dom σβ = ran σβ = Nn ,
and hence dom(σα σβ ) = ran(σα σβ ) = Nn . Since ρα , λα , σα , ρβ , λβ , σβ are partial bijection of Nn and
dom(αβ) = dom(ραβ σα σβ λαβ ), the equality αβ = ραβ σα σβ λαβ implies that σαβ = σα σβ .
Next we shall show that the equality αβ = ραβ σα σβ λαβ holds. We observe that for any z ∈ dom(αβ)
there exists a unique p ∈ Nn ∪ {(0, . . . , 0), (1, . . . , 0), . . . , (0, . . . , 1)} such that
z = (max{y, u} − y)σα−1 + x + p.

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OLEG GUTIK AND TARAS MOKRYTSKYI

Then we have that

(z) αβ = (max{y, u} − y)σα−1 + x + p αβ =

= (max{y, u} − y)σα−1 + x + p ρα σα λα β =

= (max{y, u} − y)σα−1 + p + 1 σα λα β =
= (max{y, u} − y + (p)σα + 1) λα β =

= (max{y, u} + (p)σα ) ρβ σβ λβ =
= (max{y, u} − u + (p)σα + 1) σβ λβ =
= ((max{y, u} − u)σβ + ((p)σα )σβ + 1) λβ =
= (max{y, u} − u)σβ + ((p)σα )σβ + v
and

(z)ραβ σα σβ λαβ = (max{y, u} − y)σα−1 + x + p ραβ σα σβ λαβ =
= (p + 1) σα σβ λαβ =

= (((p)σα )σβ + 1) λαβ =
= (max{y, u} − u)σβ + v + ((p)σα )σβ ).
This completes the proof of the lemma.



Proposition 2.16. Let α and β be elements of the semigroup IPF (Nn ) such that dom α = ↑x,
ran α = ↑y, dom β = ↑u and ran β = ↑v. Then the following statements hold:
(i) α is an idempotent of IPF (Nn ) if and only if λα is an inverse partial map to ρα , i.e., x = y,
and σα is the identity element of the group Sn ;
(ii) α is inverse of β in IPF (Nn ) if and only if x = v, y = u (i.e., λα is an inverse partial map
to ρβ and λβ is an inverse partial map to ρα ) and σα is inverse of σβ in the group Sn .
Proof. (i) Suppose that α is an idempotent of IPF (Nn ). Since α is an identity map of a some principal
filter of (Nn , 6), we have that x = y and hence λα is the converse partial map to ρα . Then the equalities
ρα σα λα = α = αα = ρα σα λα ρα σα λα = ρα σα σα λα
and Lemma 2.14 imply that σα = σα σα , and hence σα is the identity element of the group Sn .
The converse statement is obvious.
(ii) Suppose that α and β are inverse elements in IPF (Nn ). Then dom α = ran β and ran α = dom β
and hence we get that x = v and y = u. This and Lemma 2.14 imply that λα is the converse partial
map to ρβ and λβ is an inverse partial map to ρα . Since αβ is an idempotent of IPF (Nn ) the above
arguments imply that
αβ = ρα σα λα ρβ σβ λβ = ρα σα σβ λβ ,
and hence by statement (i) the element σα σβ is the identity of the group Sn . This implies that σα is
inverse of σβ in Sn .
The converse statement is obvious.

Remark 2.17. In the bicyclic semigroup C (p, q) the semigroup operation is determined in the following
way:
 i j−k+l
, if j > k;
 pq
i j
k l
i l
pq,
if j = k;
pq ·p q =
 i−j+k l
p
q , if j < k,
which is equivalent to the following formula:
pi q j · pk q l = pi+max{j,k}−j q l+max{j,k}−k .

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11

It is well known that the above implies that the bicyclic semigroup C (p, q) is isomorphic to the semigroup (S, ∗) which is determined on the square of non-negative integers N0 × N0 with the following
multiplication:
(1)

(i, j) ∗ (k, l) = (i + max{j, k} − j, l + max{j, k} − k).

Later, for an arbitrary positive integer n by C n (p, q) we shall denote the n-th direct power of (S, ∗),
i.e., C n (p, q) is the n-th power of N0 ×N0 with the point-wise semigroup operation defined by formula (1).
Also, by [x, y] we denote the ordered collection ((x1 , y1 ), . . . , (xn , yn )) of C n (p, q), where x = (x1 , . . . , xn )
and y = (y1 , . . . , yn ), and for arbitrary permutation σ : {1, . . . , n} → {1, . . . , n} we put

(x)σ = x(1)σ−1 , . . . , x(n)σ−1 .

Let Aut(C n (p, q)) be the group of automorphisms of the semigroup C n (p, q). We define a map Φ from
Sn into all selfmaps of the semigroup C n (p, q) putting σ 7→ Φσ , where the map Φσ : C n (p, q) → C n (p, q)
is defined by the formula:
(2)

([x, y])Φσ = [(x)σ, (y)σ].

It is obvious that the map Φσ is a bijection of C n (p, q) and Φσ1 6= Φσ2 for distinct σ1 , σ2 ∈ Sn .
Since
([x, y] ∗ [u, v])Φσ = ([max{y, u} − y + x, max{y, u} − u + v]) Φσ =
= [(max{y, u} − y + x)σ, (max{y, u} − u + v)σ] =
= [max{(y)σ, (u)σ} − (y)σ + (x)σ, max{(y)σ, (u)σ} − (u)σ + (v)σ] =
= [(x)σ, (y)σ] ∗ [(u)σ, (v)σ] =
= [x, y] Φσ ∗ [u, v] Φσ
and
([x, y])Φσ1 σ2 = [(x)σ1 σ2 , (y)σ1 σ2 ] = ([(x)σ1 , (y)σ1 ])Φσ2 = ([x, y])Φσ1 Φσ2
for any [x, y], [u, v] ∈ C n (p, q) and any σ, σ1 , σ2 ∈ Sn , the following proposition holds:
Proposition 2.18. For arbitrary positive integer n the map Φ is an injective homomorphism from Sn
into the group Aut(C n (p, q)) of automorphisms of the semigroup C n (p, q).
Theorem 2.19. For arbitrary positive integer n the semigroup IPF (Nn ) is isomorphic to the semidirect product Sn ⋉Φ C n (p, q) of the semigroup C n (p, q) by the group Sn .
Proof. We define the map Ψ : IPF (Nn ) → Sn ⋉Φ C n (p, q) in the following way:
(α)Ψ = (σα , [(x)σα , y]),
for α ∈ IPF (Nn ), where ↑x = dom α and ↑y = ran α. Since σα is a bijection, we have that Ψ is a
bijection as well.
For any α, β ∈ IPF (Nn ) with dom α = ↑x, ran α = ↑y, dom β = ↑u, ran β = ↑v by Lemma 2.15 we
have that

(αβ)Ψ = σα σβ , [((max{y, u} − y)σα−1 + x)σα σβ , (max{y, u} − u)σβ + v] =
= (σα σβ , [max{(y)σβ , (u)σβ } − (y)σβ + (x)σα σβ , max{(y)σβ , (u)σβ } − (u)σβ + v]) =
= (σα σβ , ([(x)σα , y])σβ ∗ [(u)σβ , v]) =
= (σα , [(x)σα , y])(σβ , [(u)σβ , v]) =
= (α)Ψ(β)Ψ,
and hence Ψ is an isomorphism.



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OLEG GUTIK AND TARAS MOKRYTSKYI

Every inverse semigroup S admits the least group congruence Cmg (see [33, Section III]):
sCmg t if and only if

there exists an idempotent e ∈ S

such that se = te.

Later, for any α ∈ IPF (Nn ) put (σα , [(xα )σα , yα ]) = (α)Ψ is the image of the element α by the
isomorphism Ψ : IPF (Nn ) → Sn ⋉Φ C n (p, q) which is defined in the proof of Theorem 2.19.
The following theorem describes the least group congruence on the semigroup IPF (Nn ).
Theorem 2.20. Let n be an arbitrary positive integer. Then αCmg β in the semigroup IPF (Nn ) if
and only if
σα = σβ
and
(xα )σα − yα = (xβ )σβ − yβ .
Proof. First we observe that if ε is an idempotent in IPF (Nn ) then Proposition 2.16(i) and the
definition of the map Ψ : IPF (Nn ) → Sn ⋉Φ C n (p, q) imply that σε is the identity permutation and
xε = yε . Then the following calculations
(σα , [(xα )σα , yα ]) (σε , [(xε )σε , xε ]) =
= (σα σε , [max{(yα )σε , xε } − (yα )σε + ((xα )σα )σε , max{(yα )σε , xε } − xε + xε ]) =
= (σα , [max{yα , xε } − yα + (xα )σα , max{yα , xε }]) ,
(σβ , [(xβ )σβ , yβ ]) (σε , [(xε )σε , xε ]) =
= (σβ σε , [max{(yβ )σε , xε } − (yβ )σε + ((xβ )σβ )σε , max{(yβ )σε , xε } − xε + xε ]) =
= (σβ , [max{yβ , xε } − yβ + (xβ )σβ , max{yβ , xε }]) ,
imply that for the idempotent ε ∈ IPF (Nn ) the equality αε = βε holds if and only if
σα = σβ

and

(xα )σα − yα = (xβ )σβ − yβ .

This completes the proof of the theorem.



For any positive integer n, an arbitrary permutation σ : {1, . . . , n} → {1, . . . , n} and an ordered
collection z = (z1 , . . . , zn ) of integers we put

(z)σ = z(1)σ−1 , . . . , z(n)σ−1 .

Let Aut(Zn ) be the group of automorphisms of the direct n-the power of the additive group of integers
Z(+). Next we define a map Θ : Sn → Aut(Zn ) in the following way. We put (σ)Θ = Θσ is the map
from Zn into Zn which is defined by the formula
(z)Θσ = (z)σ.
It is obvious that the maps Θ and Θσ are injective. Since
(z + v)σ = (z)σ + (v)σ
and
(z)Θσ1 σ2 = (z)(σ1 σ2 ) = ((z)σ1 )σ2 = ((z)Θσ1 )Θσ2 ,

for any z = (z1 , . . . , zn ) and v = (v1 , . . . , vn ) from the direct n-th power of the group Z(+) and any
σ, σ1 , σ2 ∈ Sn , we have that so defined map Θ : Sn → Aut(Zn ) is an injective homomorphism.
Theorem 2.21. For an arbitrary positive integer n the quotient semigroup IPF (Nn )/Cmg is isomorphic to the semidirect product Sn ⋉Θ (Z(+))n of the direct n-th power of the additive group of integers
(Z(+))n by the group of permutation Sn .
Proof. We define a map Υ : IPF (Nn ) → Sn ⋉Θ (Z(+))n in the following way. If (σα , [(xα )σα , yα ]) =
(α)Ψ is the image of α ∈ IPF (Nn ) under the isomorphism Ψ : IPF (Nn ) → Sn ⋉Φ C n (p, q) which is
defined in the proof of Theorem 2.19, then we put (α)Υ = (σα , (xα )σα − yα ).

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13

For any α, β ∈ IPF (Nn ) with dom α = ↑xα , ran α = ↑yα , dom β = ↑xβ , ran β = ↑yβ by Lemma 2.15
we have that

(αβ)Υ = σα σβ , ((max{yα , xβ } − yα )σα−1 + xα )σα σβ − (max{yα , xβ } − xβ )σβ − yβ =
= (σα σβ , max{(yα )σβ , (xβ )σβ } − (yα )σβ + (xα )σα σβ − max{(yα )σβ , (xβ )σβ } + (xβ )σβ − yβ ) =

= (σα σβ , (xα )σα σβ − (yα )σβ + (xβ )σβ − yβ ) =
= (σα σβ , ((xα )σα − yα )σβ + ((xβ )σβ − yβ )) =
= (σα , (xα )σα − yα ) · (σβ , (xβ )σβ − yβ ) =
= (α)Υ · (β)Υ,
and hence Υ is a homomorphism. It is obvious that the map Υ : IPF (Nn ) → Sn ⋉Θ (Z(+))n is
surjective. Also, Theorem 2.20 implies that αCmg β in IPF (Nn ) if and only if (α)Υ = (β)Υ. This
implies that the homomorphism Υ generates the congruences Cmg on IPF (Nn ).

Every inverse semigroup S admits a partial order:
a4b

if and only if there exists

e ∈ E(S) such that a = be.

So defined order is called the natural partial order on S. We observe that a 4 b in an inverse semigroup
S if and only if a = f b for some f ∈ E(S) (see [29, Lemma 1.4.6]).
If ε is an idempotent in IPF (Nn ) then Proposition 2.16(i) and the definition of the isomorphism
Ψ : IPF (Nn ) → Sn ⋉Φ C n (p, q) imply that σε is the identity permutation and xε = yε . Then for any
α ∈ IPF (Nn ) with (σα , [(xα )σα , yα ]) = (α)Ψ we have that
(σα , [(xα )σα , yα ]) (σε , [xε ,xε ]) =
(3)

= (σα σε , [max{(yα )σε , xε } − (yα )σε + ((xα )σα )σε , max{(yα )σε , xε } − xε + xε ]) =
= (σα , [max{yα , xε } − yα + (xα )σα , max{yα , xε }]) ,

This implies the following proposition, which describes the natural partial order on the semigroup
IPF (Nn ).
Proposition 2.22. Let n be an arbitrary positive integer and α, β ∈ IPF (Nn ). Then the following
conditions are equivalent:
(i) α 4 β;
(ii) σα = σβ , (xα )σα − yα = (xβ )σβ − yβ and xα 6 xβ in the poset (Nn , 6);
(iii) σα = σβ , (xα )σα − yα = (xβ )σβ − yβ and yα 6 yβ in the poset (Nn , 6).
An inverse semigroup S is said to be E-unitary if ae ∈ E(S) for some e ∈ E(S) implies that a ∈ E(S)
[29]. E-unitary inverse semigroups were introduced by Siatô in [35], where they were called “proper
ordered inverse semigroups”.
Formula (3) implies that if the element (σα , [(xα )σα , yα ]) · (σε , [xε ,xε ]) is an idempotent in the semidirect product Sn ⋉Φ C n (p, q) then so is (σα , [(xα )σα , yα ]). This implies the following corollary:
Corollary 2.23. For an arbitrary positive integer n the inverse semigroup IPF (Nn ) is E-unitary.
An inverse semigroup S is called F -inverse, if the Cmg -class sCmg of each element s has the top (biggest)
element with the respect to the natural partial order on S [30].
Proposition 2.24. For an arbitrary positive integer n the semigroup IPF (Nn ) is an F -inverse semigroup.
Proof. Fix an arbitrary element β0 = (σ, [x, y]) ∈ Sn ⋉Φ C n (p, q), where x = (x1 , . . . , xn ) and y =
(y1 , . . . , yn ). For every positive integer k we put βk = (σ, [x−k, y−k]), where x−k = (x1 −k, . . . , xn −k)
and y − k = (y1 − k, . . . , yn − k). It is obvious that there exists a (biggest) positive integer k0 such that
(x1 − k0 , . . . , xn − k0 ), (y1 − k0 , . . . , yn − k0 ) ∈ Nn

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OLEG GUTIK AND TARAS MOKRYTSKYI

but
(x1 − k0 − 1, . . . , xn − k0 − 1) ∈
/ Nn
or
(y1 − k0 , . . . , yn − k0 ) ∈
/ Nn .
Then Theorem 2.20 and Proposition 2.22 imply that the element βk0 is the biggest element in the
Cmg -class of the element β0 in Sn ⋉Φ C n (p, q).

Proposition 2.25. For every α, β ∈ IPF (Nn ), both sets
{χ ∈ IPF (Nn ) : α · χ = β}

and

{χ ∈ IPF (Nn ) : χ · α = β}

are finite. Consequently, every right translation and every left translation by an element of the semigroup
IPF (Nn ) is a finite-to-one map.
Proof. We shall show that the set A = {χ ∈ IPF (Nn ) : χ · α = β}. The proof of the statement that
the set {χ ∈ IPF (Nn ) : α · χ = β} is finite, is similar.
It is obvious that A is a subset of the set B = {χ ∈ IPF (Nn ) : χ · αα−1 = βα−1}. Then B is a
subset of C = {ξ ∈ IPF (Nn ) : ξ 4 βα−1 }. Since every principal ideal in the poset (Nn , 6) is finite,
Proposition 2.25 implies that C is finite, and hence so is A.

3. On a semitopological semigroup IPF (Nn )
The following theorem generalizes the Bertman-West result from [8] (and hence Eberhart-Selden
result from [12]) for the semigroup IPF (Nn ).
Theorem 3.1. For any positive integer n every Hausdorff shift-continuous topology on the semigroup
IPF (Nn ) is discrete.
Proof. Fix an arbitrary shift continuous Hausdorff topology τ on IPF (Nn ). Since for every idempotent
ε ∈ IPF (Nn ) the left and right shifts lε : IPF (Nn ) → IPF (Nn ) : x 7→ ε · x and rε : IPF (Nn ) →
IPF (Nn ) : x 7→ x · ε are continuous maps, the Hausdorffness of (IPF (Nn ), τ ) and [13, 1.5.c] implies
that the principal ideals εIPF (Nn ) and IPF (Nn )ε are closed subsets in (IPF (Nn ), τ ).
For any positive integer i = 1, . . . , n put εi is the identity map of the subset ↑(1, . . . , |{z}
2 , . . . , 1) of

the poset (Nn , 6). It is clear that

i-th

H(I) = IPF (Nn ) \ (ε1 IPF (N)n ∪ · · · εn IPF (Nn ) ∪ IPF (Nn )ε1 ∪ · · · ∪ IPF (Nn )εn ) .
Then above part of the proof implies that H(I) is an open subset of (IPF (Nn ), τ ), and by Theorem 2.5,
H(I) is a finite discrete open subspace of (IPF (Nn ), τ ).
Since By Proposition 2.1(vii) the semigroup IPF (Nn ) is simple, for an arbitrary α ∈ IPF (Nn )
there exist β, γ ∈ IPF (Nn ) such that βαγ = I. Theny Proposition 2.25 implies that the point α has
a finite open neighbourhood in (IPF (Nn ), τ ), and hence α is an isolated point in (IPF (Nn ), τ ). 
The following theorem generalizes Theorem I.3 from [12].
Theorem 3.2. If for some positive integer n the semigroup IPF (Nn ) is dense in a Hausdorff semitopological semigroup (S, ·) and I = SIPF (Nn ) 6= ∅, then I is a two-sided ideal in S.
Proof. Fix an arbitrary element y ∈ I. If x · y = z ∈
/ I for some x ∈ IPF (Nn ) then there exists an
open neighbourhood U(y) of the point y in the space S such that {x} · U(y) = {z} ⊂ IPF (Nn ). Then
the open neighbourhood U(y) contains infinitely many elements of the semigroup IPF (Nn ) which
contradicts Proposition 2.25. The obtained contradiction implies that x · y ∈ I for all x ∈ IPF (Nn )
and y ∈ I. The proof of the statement that y · x ∈ I for all x ∈ IPF (Nn ) and y ∈ I is similar.
Suppose to the contrary that x · y = w ∈
/ I for some x, y ∈ I. Then w ∈ IPF (Nn ) and the separate
continuity of the semigroup operation in S implies that there exist open neighbourhoods U(x) and U(y)
of the points x and y in S, respectively, such that {x} · U(y) = {w} and U(x) · {y} = {w}. Since both
neighbourhoods U(x) and U(y) contain infinitely many elements of the semigroup IPF (Nn ), both

THE MONOID OF ORDER ISOMORPHISMS OF PRINCIPAL FILTERS ...

15

equalities {x} · U(y) = {w} and U(x) · {y} = {w} contradict the mentioned above part of the proof,
because {x} · (U(y) ∩ IPF (Nn )) ⊆ I. The obtained contradiction implies that x · y ∈ I.

We recall that a topological space X is said to be:
• compact if every open cover of X contains a finite subcover;
• countably compact if each closed discrete subspace of X is finite;
• feebly compact if each locally finite open cover of X is finite;
• pseudocompact if X is Tychonoff and each continuous real-valued function on X is bounded.
According to Theorem 3.10.22 of [13], a Tychonoff topological space X is feebly compact if and only if X
is pseudocompact. Also, a Hausdorff topological space X is feebly compact if and only if every locally
finite family of non-empty open subsets of X is finite. Every compact space and every sequentially
compact space are countably compact, every countably compact space is feebly compact (see [3]).
A topological semigroup S is called Γ-compact if for every x ∈ S the closure of the set {x, x2 , x3 , . . .} is
compact in S (see [26]). Since by Proposition 2.7 for every positive integer n the semigroup IPF (Nn )
contains the bicyclic semigroup as a subsemigroup the results obtained in [2], [4], [5], [22], [26] imply
the following corollary
Corollary 3.3. Let n be an arbitrary non-negative integer. If a Hausdorff topological semigroup S
satisfies one of the following conditions:
(i) S is compact;
(ii) S is Γ-compact;
(iii) S is a countably compact topological inverse semigroup;
(iv) the square S × S is countably compact; or
(v) the square S × S is a Tychonoff pseudocompact space,
then S does not contain the semigroup IPF (Nn ).
Also, Theorems 2.12, 2.21 and Corollary 3.3 imply the following corollary:
Corollary 3.4. Let n be an arbitrary non-negative integer. If a Hausdorff topological semigroup S
satisfies one of the following conditions:
(i) S is compact;
(ii) S is Γ-compact;
(iii) S is a countably compact topological inverse semigroup;
(iv) the square S × S is countably compact; or
(v) the square S × S is a Tychonoff pseudocompact space,
then for every homomorphism h : IPF (Nn ) → S the image (IPF (Nn ))h is a subgroup of S. Moreover,
for every homomorphism h : IPF (Nn ) → S there exists a unique homomorphism uh : Sn ⋉Θ (Z(+))n →
S such that the following diagram
IPF (Nn )
Υ



h

// S
✉::
✉
✉
✉✉
✉✉
✉
✉
✉✉
✉✉uh
✉
✉
✉✉
✉✉
✉
✉

Sn ⋉Θ (Z(+))n
commutes.
Proposition 3.5. Let n be an arbitrary positive integer. Let S be a Hausdorff topological semigroup
which contains a dense subsemigroup IPF (Nn ). Then for every c ∈ IPF (Nn ) the set
Dc = {(x, y) ∈ IPF (Nn ) × IPF (Nn ) : x · y = c}

16

OLEG GUTIK AND TARAS MOKRYTSKYI

is an open-and-closed subset of S × S.
Proof. By Theorem 3.1, IPF (Nn ) is a discrete subspace of S and hence Theorem 3.3.9 of [13] implies
that IPF (Nn ) is an open subspace of S. Then the continuity of the semigroup operation of S implies
that Dc is an open subset of S × S for every c ∈ IPF (Nn ).
Suppose that there exists c ∈ IPF (Nn ) such that Dc is a non-closed subset of S × S. Then there
exists an accumulation point (a, b) ∈ S × S of the set Dc . The continuity of the semigroup operation
in S implies that a · b = c. But IPF (Nn ) × IPF (Nn ) is a discrete subspace of S × S and hence by
Theorem 3.2 the points a and b belong to the two-sided ideal I = S \ IPF (Nn ). This implies that the
product a · b ∈ S \ IPF (Nn ) cannot be equal to the element c.

Theorem 3.6. Let n be an arbitrary positive integer. If a Hausdorff topological semigroup S contains
IPF (Nn ) as a dense subsemigroup then the square S × S is not feebly compact.
Proof. By Proposition 3.5 for every c ∈ IPF (Nn ) the square S ×S contains an open-and-closed discrete
subspace Dc . In the case when c is the unit I of IPF (Nn ), by Corollary 2.8 there exists a subsemigroup
C of IPF (Nn ) which is isomorphic to the bicyclic monoid C (p, q) and contains I. If we identify the
elements of C with the elements the bicyclic monoid C (p, q) by an isomorphism h : C (p, q) → C then
the subspace Dc contains an infinite subset {(h(q i ), h(pi )) : i ∈ N0 } and hence Dc is infinite. This implies
that the square S × S is not feebly compact.

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Faculty of Mathematics, National University of Lviv, Universytetska 1, Lviv, 79000, Ukraine
E-mail address: ogutik@gmail.com, ovgutik@yahoo.com, tmokrytskyi@gmail.com

