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Article

A Semigroup Is Finite Iff It Is Chain-Finite and Antichain-Finite

1
Pidstryhach Institute for Applied Problems of Mechanics and Mathematics, National Academy of Sciences of Ukraine, Naukova 3b, 79060 Lviv, Ukraine
2
Faculty of Mehcanics and Mathematics, Ivan Franko National University of Lviv, Universytetska 1, 79000 Lviv, Ukraine
3
Katedra Matematyki, Jan Kochanowski University in Kielce, Universytecka 7, 25-406 Kielce, Poland
4
Kurt Gödel Research Center, Institute of Mathematics, University of Vienna, 1090 Vienna, Austria
*
Author to whom correspondence should be addressed.
These authors contributed equally to this work.
Submission received: 8 December 2020 / Revised: 10 January 2021 / Accepted: 13 January 2021 / Published: 16 January 2021
(This article belongs to the Special Issue Topological Algebra)

Abstract

:
A subset A of a semigroup S is called a chain (antichain) if a b { a , b } ( a b { a , b } ) for any (distinct) elements a , b A . A semigroup S is called periodic if for every element x S there exists n N such that x n is an idempotent. A semigroup S is called (anti)chain-finite if S contains no infinite (anti)chains. We prove that each antichain-finite semigroup S is periodic and for every idempotent e of S the set e = { x S : n N ( x n = e ) } is finite. This property of antichain-finite semigroups is used to prove that a semigroup is finite if and only if it is chain-finite and antichain-finite. Furthermore, we present an example of an antichain-finite semilattice that is not a union of finitely many chains.
MSC:
20M10; 06F05; 05E16

1. Introduction

It is well-known that a partially ordered set X is finite iff all chains and antichains in X are finite. The notions of chain and antichain are well-known in the theory of order (see, e.g., ([1] (O-1.6)) or [2]). In this paper we present a similar characterization of finite semigroups in terms of finite chains and antichains.
Let us recall that a magma is a set S endowed with a binary operation S × S S , x , y x y . If the binary operation is associative, then the magma S is called a semigroup. A semilattice is a commutative semigroup whose elements are idempotents. Each semilattice S carries a natural partial order ≤ defined by x y iff x y = y x = x . Observe that two elements x , y of a semilattice are comparable with respect to the partial order ≤ if and only if x y { x , y } . This observation motivates the following algebraic definition of chains and antichains in any magma.
A subset A of a magma S is defined to be
  • a chain if x y { x , y } for any elements x , y A ;
  • an antichain if x y { x , y } for any distinct elements x , y A .
The definition implies that each chain consists of idempotents.
A magma S is defined to be (anti)chain-finite if it contains no infinite (anti)chains.
Let us note that chain-finite semilattices play an important role in the theory of complete topological semigroups. In [3], Stepp showed that for each homomorphism f : X Y from a chain-finite semilattice X to a Hausdorff topological semigroup Y, the image f [ X ] is closed in Y. Banakh and Bardyla [4] extended the result of Stepp to the following characterization:
Theorem 1.
For a semilattice X the following conditions are equivalent:
  • X is chain-finite;
  • X is closed in each Hausdorff topological semigroup containing X as a discrete subsemigroup;
  • For each homomorphism f : X Y into a Hausdorff topological semigroup Y, the image f [ X ] is closed;
For other completeness properties of chain-finite semilattices see [4,5,6]. Antichain-finite posets and semilattices were investigated by Yokoyama [7].
The principal result of this note is the following theorem characterizing finite semigroups.
Theorem 2.
A semigroup S is finite if and only if it is chain-finite and antichain-finite.
A crucial step in the proof of this theorem is the following proposition describing the (periodic) structure of antichain-finite semigroups.
A semigroup S is called periodic if for every x S there exists n N such that x n is an idempotent of S. In this case
S = e E ( S ) e ,
where E ( S ) = { x S : x x = x } is the set of idempotents of S and
e = { x S : n N ( x n = e ) }
for e E ( S ) .
Proposition 1.
Each antichain-finite semigroup S is periodic and for every e E ( S ) the set e is finite.
Theorem 2 and Proposition 1 will be proved in the next section.
Remark 1.
Theorem 2 does not generalize to magmas. To see this, consider the set of positive integers N endowed with the following binary operation: n m = n if n < m and n m = 1 if n m . This magma is infinite but each nonempty chain in the magma is of the form { 1 , n } for some n N , and each nonempty antichain in this magma is a singleton.
Next we present a simple example of an antichain-finite semilattice which is not a union of finitely many chains.
Example 1.
Consider the set
S = { 2 n 1 , 0 : n N } { 2 n , m : n , m N , m 2 n }
endowed with the semilattice binary operation
x , i · y , j = x , i i f   x = y   a n d   i = j ; x 1 , 0 i f   x = y   a n d   i j ; x , i i f   x < y ; y , j i f   y < x .
It is straightforward to check that the semilattice S has the following properties:
1.
S is antichain-finite;
2.
S has arbitrarily long finite antichains;
3.
S is not a union of finitely many chains;
4.
The subsemilattice L = { 2 n 1 , 0 : n N } of S is a chain;
5.
S admits a homomorphism r : S L such that r 1 ( x , 0 ) = { y , i S : y { x , x + 1 } } is finite for every element x , 0 L .
Example 1 motivates the following question.
Question 1.
Let S be an antichain-finite semilattice. Is there a finite-to-one homomorphism r : S Y to a semilattice Y which is a finite union of chains?
A function f : X Y is called finite-to-one if for every y Y the preimage f 1 ( y ) is finite.

2. Proofs of the Main Results

In this section, we prove some lemmas implying Theorem 2 and Proposition 1. More precisely, Proposition 1 follows from Lemmas 1 and 4; Theorem 2 follows from Lemma 5.
The following lemma exploit ideas of Theorem 1.9 from [8].
Lemma 1.
Every antichain-finite semigroup S is periodic.
Proof. 
Given any element x S we should find a natural number n N such that x n is an idempotent. First we show that x n = x m for some n m . Assuming that x n x m for any distinct numbers n , m , we conclude that the set A = { x n : n N } is infinite and for any n , m N we have x n x m = x n + m { x n , x m } , which means that A is an infinite antichain in S. However, such an antichain cannot exist as S is antichain-finite. This contradiction shows that x n = x m for some numbers n < m and then for the number k = m n we have x n + k = x m = x n . By induction we can prove that x n + p k = x n for every p N . Choose any numbers r , p N such that r + n = p k and observe that
x r + n x r + n = x r + n x p k = x r x n + p k = x r x n = x r + n ,
which means that x r + n is an idempotent and hence S is periodic. □
An element 1 S is called an identity of S if x 1 = x = 1 x for all x S . For a semigroup S let S 1 = S { 1 } where 1 is an element such that x 1 = x = 1 x for every x S 1 . If S contains an identity, then we will assume that 1 is the identity of S and hence S 1 = S .
For a set A S and element x S we put
x A = { x a : a A } and A x = { a x : a A } .
For any element x of a semigroup S, the set
H x = { y S : y S 1 = x S 1 S 1 y = S 1 x }
is called the H -class of x. By Lemma I.7.9 [9], for every idempotent e its H -class H e coincides with the maximal subgroup of S that contains the idempotent e.
Lemma 2.
If a semigroup S is antichain-finite, then for every idempotent e of S its H -class H e is finite.
Proof. 
Observe that the set H e { e } is an antichain (this follows from the fact that the left and right shifts in the group H e are injective). Since S is antichain-finite, the antichain H e { e } is finite and so is the set H e . □
Lemma 3.
If a semigroup S is antichain-finite, then for every idempotent e in S we have
( H e · e ) ( e · H e ) H .
Proof. 
Given any elements x e and y H e , we have to show that x y H e and y x H e . Since x e , there exists a number n N such that x n = e . Then x n + 1 S 1 = e x S 1 e S 1 and e S 1 = x 2 n S 1 x n + 1 S 1 , and hence e S 1 = x n + 1 S . By analogy we can prove that S 1 e = S 1 x n + 1 . Therefore, x n + 1 H e .
Then x y = x ( e y ) = ( x e ) y = ( x x n ) y = x n + 1 y H e and y x = ( y e ) x = y ( e x ) = y x n + 1 H e . □
For each k N by [ N ] k we denote the set of all k-element subsets of N . The proofs of the next two lemmas essentially use the classical Ramsey Theorem, so let us recall its formulation, see ([10] (p. 16)) for more details.
Theorem 3
(Ramsey). For any n , k N and map χ : [ N ] k n = { 0 , , n 1 } there exists an infinite subset I N such that χ [ I ] k = { c } for some number c n .
Lemma 4.
If a semigroup S is antichain-finite, then for every idempotent e E ( S ) the set e is finite.
Proof. 
By Lemma 2, the H -class H e is finite. Assuming that e is infinite, we can choose a sequence ( x n ) n ω of pairwise distinct points of the infinite set e H e .
Let P = { n , m ω × ω : n < m } and χ : P 5 = { 0 , 1 , 2 , 3 , 4 } be the function defined by
χ ( n , m ) = 0 if   x n x m = x n ; 1 if   x m x n = x n ; 2 if   x n x m = x m ; 3 if   x m x n = x m ; 4 otherwise .
By the Ramsey Theorem 3, there exists an infinite subset Ω ω such that χ [ P ( Ω × Ω ) ] = { c } for some c { 0 , 1 , 2 , 3 , 4 } .
If c = 0 , then x n x m = x n for any numbers n < m in Ω . Fix any two numbers n < m in Ω . By induction we can prove that x n x m p = x n for every p N . Since x m e , there exists p N such that x m p = e . Then x n = x n x m p = x n e H e by Lemma 3. However, this contradicts the choice of x n .
By analogy we can derive a contradiction in cases c { 1 , 2 , 3 } .
If c = 4 , then the set A = { x n } n Ω is an infinite antichain in S, which is not possible as the semigroup S is antichain-finite.
Therefore, in all five cases we obtain a contradiction, which implies that the set e is finite. □
Our final lemma implies the non-trivial “if” part of Theorem 2.
Lemma 5.
A semigroup S is finite if it is chain-finite and antichain-finite.
Proof. 
Assume that S is both chain-finite and antichain-finite. By Lemma 1, the semigroup S is periodic and hence S = e E ( S ) e . By Lemma 4, for every idempotent e E ( S ) the set e is finite. Now it suffices to prove that the set E ( S ) is finite.
To derive a contradiction, assume that E ( S ) is infinite and choose a sequence of pairwise distinct idempotents ( e n ) n ω in S. Let P = { n , m ω × ω : n < m } and χ : P { 0 , 1 , 2 , 3 , 4 , 5 } be the function defined by the formula
χ ( n , m ) = 0 if   e n e m { e n , e m }   and   e m e n { e n , e m } ; 1 if   e n e m = e n   and   e m e n { e n , e m } ; 2 if   e n e m = e m   and   e m e n { e n , e m } ; 3 if   e n e m { e n , e m }   and   e m e n = e n ; 4 if e n e m { e n , e m }   and   e m e n = e m ; 5 if   e n e m { e n , e m }   and   e m e n { e n , e m } .
The Ramsey Theorem 3 yields an infinite subset Ω ω such that χ [ P ( Ω × Ω ) ] = { c } for some c { 0 , 1 , 2 , 3 , 4 , 5 } .
Depending on the value of c, we shall consider six cases.
If c = 0 (resp. c = 5 ), then { e n } n ω is an infinite (anti)chain in S, which is forbidden by our assumption.
Next, assume that c = 1 . Then e n e m = e n and e m e n { e n , e m } for any numbers n < m in Ω . For any number k Ω , consider the set Z k = { e n e k : k < n Ω } . Observe that for any e n e k , e m e k Z k we have
( e n e k ) ( e m e k ) = e n ( e k e m ) e k = e n e k e k = e n e k ,
which means that Z k is a chain. Since S is chain-finite, the chain Z k is finite.
By induction we can construct a sequence of points ( z k ) k ω k ω Z k and a decreasing sequence of infinite sets ( Ω k ) k ω such that Ω 0 Ω and for every k ω and n Ω k we have e n e k = z k and n > k . Choose an increasing sequence of numbers ( k i ) i ω such that k 0 Ω 0 and k i Ω k i 1 for every i N . We claim that the set Z = { z k i : i ω } is a chain. Take any numbers i , j ω and choose any number n Ω k i Ω k j .
If i j , then
z k i z k j = ( e n e k i ) ( e n e k j ) = e n ( e k i e n ) e k j = e n e k i e k j = e n e k i = z k i .
If i > j , then k i Ω k i 1 Ω k j and hence
z k i z k j = ( e n e k i ) ( e n e k j ) = e n ( e k i e n ) e k j = e n ( e k i e k j ) = e n z k j = e n ( e n e k j ) = e n e k j = z k j .
In both cases we obtain that z k i z k j { z k i , z k j } , which means that the set Z = { z k i : i ω } is a chain. Since S is chain-finite, the set Z is finite. Consequently, there exists z Z such that the set Λ = { i ω : z k i = z } is infinite. Choose any numbers i < j in the set Λ and then choose any number n Ω k j Ω k i . Observe that k j Ω k j 1 Ω k i and hence e k j e k i = z k i = z . Then
e k j = e k j e k j = ( e k j e n ) e k j = e k j ( e n e k j ) = e k j z k j = e k j z = e k j z k i = e k j ( e k j e k i ) = e k j e k i { e k j , e k j }
as c = 1 .
By analogy we can prove that the assumption c { 2 , 3 , 4 } also leads to a contradiction. □

Author Contributions

The authors contributed equally to this work. All authors have read and agreed to the published version of the manuscript.

Funding

The third author was supported by the Austrian Science Fund FWF (Grant M 2967).

Institutional Review Board Statement

Not applicable.

Informed Consent Statement

Not applicable.

Data Availability Statement

This research did not report any data.

Conflicts of Interest

The authors declare no conflict of interest.

References

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Banakh, I.; Banakh, T.; Bardyla, S. A Semigroup Is Finite Iff It Is Chain-Finite and Antichain-Finite. Axioms 2021, 10, 9. https://doi.org/10.3390/axioms10010009

AMA Style

Banakh I, Banakh T, Bardyla S. A Semigroup Is Finite Iff It Is Chain-Finite and Antichain-Finite. Axioms. 2021; 10(1):9. https://doi.org/10.3390/axioms10010009

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Banakh, Iryna, Taras Banakh, and Serhii Bardyla. 2021. "A Semigroup Is Finite Iff It Is Chain-Finite and Antichain-Finite" Axioms 10, no. 1: 9. https://doi.org/10.3390/axioms10010009

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