On some semigroups generated from Cayley functions

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J. Semigroup Theory Appl. 2018, 2018:5 https://doi.org/10.28919/jsta/3562 ISSN: 2051-2937

ON SOME SEMIGROUPS GENERATED FROM CAYLEY FUNCTIONS

LEJO J. MANAVALAN∗, P.G. ROMEO

Department of Mathematics, Cochin University of Science and Technology, Cochin University P.O., 682022,

India

Copyright c2018 Manavalan and Romeo. This is an open access article distributed under the Creative Commons Attribution License, which

permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract. Any transformation on a setSis called a Cayley function onSif there exists a semigroup operation on

Ssuch thatβ is an inner-translation. In this paper we describe a method to generate a semigroup withknumber of

idempotents, study some properties of such semigroups like greens relations and bi-ordered sets.

Keywords:semigroups; idempotents; cayley functions; greens relation.

2010 AMS Subject Classification:20M99.

1. Introduction

Letα be a transformation on a set S. Following [4] we say thatα is a Cayley function onS

if there is a semigroup with universeSsuch thatα is an inner translation of the semigroupS. A

section of group theory has developed historically through the characterisation of inner

transla-tions as regular permutatransla-tions. The problem of characterising inner translatransla-tions of semigroups

was raised by Schein [7] and solved by Goralcik and Hedrlin [6].In 1972 Zupnik characterised

∗_{Corresponding author}

E-mail address: [email protected]

Received November 14, 2017

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all Cayley functions algebraically(in powers ofβ) [4]. In 2016, Araoujo et all characterised the

Cayley functions using functional digraphs[5]. In this paper we use a Cayley permutation to

generate a semigroup withnelements andk≤nidempotents and discuss some of its properties.

In the sequelβ will denote a function mapping a non-empty setSonto itself. For any positive

integern,βndenotes thenthiterate ofβ. Byβ0we mean the identity function onS, soβ0(x) =

x. LetSbe a set, thenT(S)denotes the set of all transformations fromStoS.

2. Preliminaries

A semigroup is a non empty set S along with a binary operation ∗ on S such that(S,∗) is

associative. An idempotent elementεinSis an element such thatε2=ε∗ε=ε. The set of all

idempotents inSis denoted byE(S)

Ifais an element of a semigroupS, the smallest left ideal containingaisSa∪ {a}orS1athe principal left ideal generated bya. The equivalence relationL onSis defined onSbyaL b

if and only ifS1a=S1b. Similarly we say thataRbif and only ifaS1=bS1. The following is due to J.A. Green.

Lemma 2.1. Leta,bbe elements of a semigroup S. Then

• aL bif and only if∃x,y∈S1such thatxa=bandyb=a

• aR bif and only if∃x,y∈S1such thatax=bandby=a

The following lemma is lemma 2.1 of [3]

Lemma 2.2. The relationsL andR commute and so the relationD=L◦R=R◦L is the smallest equivalence relationL∨R containing bothL andR. We defineH =L ∩R

Let S be a semigroup. For a fixed a∈S, the mapping λa:S→S [ρa :S→S] defined by

λa(x) =ax[ρa(x) =xa]is called a left [right] inner translation of S.

Definition 2.1. Letβ be a transformation on a setS. We say thatβ is a Cayley function on S

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Note that β is a left inner translation of a semigroup(S,∗) if and only ifβ is a right inner

translation of the semigroup(S, .), where for alla,b∈S,a∗b=b.a.

Definition 2.2. The stabilizer of β ∈T(S) is the smallest integers≥0 such thatimg(βs) =

img(βs+1). If such an s does not exist, we say thatβ has no stabilizer.

The following definition though originally by Zupnik was modified by Araujo in [5]

Definition 2.3. Supposeβ ∈T(S)has the stabilizers. Ifs>0, we define the subset Ω_β ofS

by:

Ω_β ={a∈S:βn(a)∈Ran(βs)if and only ifn≥s−1}

Ifs=0, we defineΩ_β to beS.

Theorem 2.1. [4] Letβ ∈T(S). Thenβ is a Cayley function if and only if exactly one of the

following conditions holds:

a: has no stabilizer and there existsa∈Ssuch thatβn(a)∈/img(βn+1)for everyn≥0;

b: has the stabilizerssuch thatβ|img(βs)is one-to-one and there existsa∈Ω_β such that

βm(a) =βn(a)impliesβm=βnfor allm,n≥0; or

c: has the stabilizerssuch thatβ|img(βs)is not one-to-one and there existsa∈Ω_β such

that:

(1) βm(a) =βn(a)impliesm=nfor allm,n≥0 ;and

(2) For everyn>s, there are pairwise distinct elementsy₁,y₂, ...ofSsuch thatβ(y1) =

βn(a),β(y_k) =y_k−1for everyk≥2, and ifn>0 theny16=βn−1(a).

A Cayley function that is also a permutation is called a Cayley permutation. Similarly a

Cayley function that also an idempotent is called a Cayley Idempotent.

3. Some Class of Semigroup from Cayley Functions

In this section we construct a semigroupS_β from a Cayley permutationβ on a finite set and

study some of its properties. LetSbe a set withnelements, anda∈Sbe a fixed element. For

anya₁,inSwe consider set{r:βr(a) =a1}of all non-negative integers such thatβr(a) =a1 in case the set is non empty we defineδai=min{r:β

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Theorem 3.1. LetSbe a set with nelements for 0<k≤nthen it is possible to construct a semigroup withkidempotents using a Cayley permutation.

Proof. Let 0<k<n, anda₁ be a fixed element ofS. Letβ be the permutation that such that

β = (a1a2a3...an−kan−k+1)ann−k+1 cycle inSnthat permutesn−k+1 terms and fixes the rest of thek−1 terms, it is a Cayley function by theorem 1 above. Now consider the binary

operation onSgiven by

ai∗aj=

  

 

βδai+1(aj) ifaiis not a fixed element ofβ a_i ifa_iis a fixed element ofβ

.

whereδai=min{r:β

r₍_a

1) =ai}. Fork=nconsider the identity permutation with the same construction. We can see that∗ is well defined binary operation and that ∗ is associative. So

(S,∗)is a semigroup. By the choice of the permutation β and the definition of∗, we can see

thata_n₋_k+1is an idempotent as

a_n₋_k+1∗a_n₋_k+1=βn−k+1(an−k+1) =an−k+1

and all thek−1 fixed elements ofβ are also idempotents.

Example 1.1. Let S={abcde} and let k =3 then we chose β = a b c d e

b c a d e

!

now

following the construction as in the above theorem we have the following Cayley table onS

* a b c d e

a b c a d e

b c a b d e

c a b c d e

d d d d d d

e e e e e e

For the rest of the paper we denote the semigroup generated in the above theorem asS_β.

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(1) Ifa,bare both non-fixed elements ofβ , thenaR b. Ifais a fixed element ofβ, thena

isR related only to itself.

(2) Ifa,bare both non-fixed elements ofβ, thenaL b. Ifa,bare both fixed elements of β thenaL b

(3) S_β contains twoDclasses.

(4) S_β containsknumber ofH classes.

Proof. (1) Ifais not a fixed element ofβ thenaS_β =S_β, ifais a fixed element ofβ then

aS_β ={a}. Hence (1). In correspondence to lemma1 we have for non-fixed elementsa_i

,aj ofSβ aq∗ai=ajwhereq+i= jmod(n−k+1)for fixed elementsai∗ai=ai

(2) Ifais not a fixed element of β thenS_βa=S_β, ifais a fixed element ofβ thenS_βa=

{b:bis a fixed element of}. And in correspondence to lemma1

(3) from lemma 2D =L◦R=R◦L. Hence from 1 and 2 we get twoDclasses

(4) from lemma 2H =L ∩R and hence kH classes

Generally the egg box picture of the semigroupS_β is as follows.

a₁,a₂,... ...a_n₋k+1

a_n₋k+2

a_n₋k+3

a_n₋_k+4

.

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Remark 3.1. Let β be ann−k+1 cycle (a Cayley Permutation) and S_β be the semigroup

constructed as in theorem 3.1, then it is easy to observe the following properties.

• {a₁,a₂, ....a_n₋_k+1}forms a subgroup ofS_β.

• a_n₋_k+1is the identity element ofS_β.

• k-1 idempotents act as left zeros ( absorbing elements ).

• Idempotents do not commute.

• S_β is a regular semigroup.

• In factS_β is a completely regular semigroup

• S_β is a not an inverse semigroup.

• S_β is a union of a group and a band.

• E(S_β)forms a sub-semigroup ofS_β( i. e,S_βis an orthodox semigroup.)

Conflict of Interests

The authors declare that there is no conflict of interests.

Acknowledgement

The first authors was financially supported by KSCSTE, Thiruvananthapuram.

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