# Tight Representations of 0-E-Unitary Inverse Semigroups

Authors: Bahman Tabatabaie Shourijeh; Asghar Jokar
Journal: Abstract and Applied Analysis
 (2011)

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## Abstract

We study the tight representation of
a semilattice in{0,1} by some examples. Then we introduce the
concept of the complex tight representation of an inverse semigroup
S by the concept of the tight representation of the semilattice of
idempotents E of S in {0,1}. Specifically we describe the tight
representation of a 0-E-unitary inverse semigroup and prove that
if σ is a tight semilattice representation of the 0-E-unitary inverse
semigroup S in {0,1}, then σ is a complex tight representation.


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## Body

## 1. Introduction

Asemigroup is a set equipped with an associative binary operation. A monoid is a semigroup with an identity. A semigroup S is said to be an inverse  semigroup, provided there exists, for each s in S, a unique element s* in S such that (1.1)s=ss*s,s*=s*ss*
Good references for inverse semigroups are [1–3].For a given setX, let I(X) be the set of all bijective functions f:A→B, where A and B are subsets of X. The multiplication on I(X) is by composition of functions, defined on the largest possible domain. More precisely, for f,g∈I(X), let fog be the function with dom⁡(fog)=g-1(ran(g)∩dom⁡(f)), and fog(x)=f(g(x)). The involution on I(X) sends a function to its inverse. I(X) is called the inverse semigroup of partial bijections on X.By the Wagner-Preston representation theorem, (see [1,1.5.1])  every inverse semigroup is an inverse semigroup of partial bijection.LetS be an inverse semigroup. An idempotent is an element e∈S such that e2=e. The set of idempotents of S is usually denoted by E(S), or just E. A partial bijection is idempotent if and only if it is the identity function on its domain.Thenatural  partial  order≤ on S is defined by (1.2)s≤tiffs=teforsomeidempotente.
The natural partial order induces a semilattice structure on the set E(S) of idempotents by the order (1.3)e≤fiffe=ef.
So, one often refers to E(S) as the semilattices of idempotents of S. For f,g in I(X),f≤g if and only if g restricted to dom⁡(f) is f.LetBn={(i,j):1≤i,j≤n}∪{0}. Define a multiplication on Bn by (1.4)(i,j)(k,l)={(i,l),i=j,0,otherwise,
and (i,j)0=0(i,j)=0. Define the involution on Bn by (i,j)*=(j,i). The inverse semigroup Bn in called a Brandt semigroup.


## 2. Tight Representations of Semilattices

In this section we define the tight representation of a semilatticeE on {0,1} and introduce two characteristic functions on E that are tight representations. One can see more about representations and semilattices in [4–7].Definition 2.1.
LetE be a partially ordered set. A subset F⊆E is said to be connected if, for every f1 and f2 in F, there exists an element f in F such that
(2.1)f≤f1,f≤f2.
A component of E is a maximal connected subset of E. For a partially ordered set E with the minimum element 0, we denote by Emin⁡ the set of all minimal elements of E*=E∖{0}.Definition 2.2.
Given a partially ordered setE with smallest element 0, we say that two elements s and t in E are disjoint, in symbols s⊥t, if there is no nonzero u∈E such that u≤s,t. Otherwise we say that s and tintersect, in symbols s∩t≠∅.
For any subsetU of E, we say that a subset V⊆U is a cover for U if, for every nonzero u∈U, there exists v∈V such that u∩v≠∅.
Asemilattice is a partially ordered set E such that for every s,t∈E, the set {u∈E:u≤s,t} contains a maximum element.
From now on we will fix a semilatticeE.Definition 2.3.
For a finite subsetF⊆E, define [0,F] to be the subset of E given by
(2.2)[0,F]={e∈E:e≤f,∀f∈F},
and denote by F⊥ the subset of E given by
(2.3)F⊥={e∈E:e⊥f,∀f∈F}.
It is obvious that 0∈[0,F] and if F is not contained in a component of E*, then [0,F]={0}. If F and G are finite subsets of E, we denote by EF,G the subset [0,F]∩G⊥ of E.Notice that ifF=G=∅, than EF,G=E, if F=∅, EF,G=G⊥ and if G=∅, EF,G=[0,F].Ife≤f, then E{e},{f}={0} and E*{e},{f}=∅. However E{f},{e} is not necessarily zero. Note that if e and f belong to different components of E*, then E{e},{f}=(0,e]. For elements e and f in E such that e≤f, e is said to be dense in f if E{f},{e}={0}.Definition 2.4.
A mapσ:E→{0,1} is said to be a representation of E in {0,1}, if σ(0)=0 and σ(x∧y)=σ(x)σ(y), for all x,y in E. We say that σ is tight if for all finite subsets F,G⊆E, and for all finite cover H for EF,G, one has that
(2.4)sgn⁡(∑h∈Hσ(h))=∏f∈Fσ(f)∏g∈G(1-σ(g)).Proposition 2.5.
Lete and f be in E with e being dense in f. Then σ(e)=σ(f) for every tight representation σ of E in {0,1}.Proof.
Suppose thatσ is a tight representation of E in {0,1} and choose e,f in E such that E{f},{e}={0}. Then ∅ is a cover for E{f},{e}. So by the definition of tight representation we have σ(f)(1-σ(e))=0. Therefore σ(f)≤σ(e). On the other hand, since e≤f, then σ(e)≤σ(f).Theorem 2.6.
LetE be a semilattice with minimum element 0. If e∈Emin⁡, then χ[e,∞) is a tight representation of E in {0,1}.Proof.
Setσ=χ[e,∞). If x,y∈E are such that x≤y, then σ(x)≤σ(y). On the other hand if x and y are disjoint, then σ(x) and σ(y) are disjoint too. So σ(x)≤1-σ(y). More generally, if F and G are finite subsets of E, and h∈E is such that h≤f for every f∈F, and h⊥g, for every g∈G, then
(2.5)σ(h)≤∏f∈Fσ(f)∏g∈G(1-σ(g)).
Conversely, let F,G be finite subsets of E, and let H be a cover for EF,G. To prove the inequality
(2.6)sgn⁡(∑h∈Hσ(h))≥∏f∈Fσ(f)∏g∈G(1-σ(g)),
we see that if the right-hand side is 0, then the inequality holds obviously. So suppose that the right-hand side is 1. Then we show that the left-hand side is 1 too. Since σ=χ[e,∞), we have F⊆[e,∞) and G∩[0,∞)=∅. Also e∈EF,G. Then there exists h∈H such that h∩e≠∅. This means that there exists a nonzero t∈E such that t≤h,e. Since e∈Emin⁡, then e≤h and so h∈[0,∞) and σ(h)=1. Therefore the left-hand side is 1 too.By the definition ofEmin⁡, one can show that every element of Emin⁡ is the minimum element of some component of E*. But it may happen that some component of E* does not have a minimum element. So the following theorem holds.Theorem 2.7.
IfF is a component of E*, then χF is a tight representation of E in {0,1}.


## 3. Complex Tight Representations of 0-E-Unitary Inverse Semigroups

The class ofE-unitary inverse semigroups is one of the most important in inverse semigroup theory. When an inverse semigroup contains a zero, then every element of E must be idempotent. Thus motivated by Szendrei [8], we define the class of 0-E-unitary inverse semigroups (although she called them E*-unitary). The term 0-E-unitary appears to be due to Meakin and Sapir [9]. More references for 0-E-unitary inverse semigroups are [10–12].Throughout this section we define complex tight representations of inverse semigroups and prove that every semilattice tight representation on a 0-E-unitary inverse semigroup is a complex tight representation.Definition 3.1.
An inverse semigroupS with semilattice of idempotent E is E-unitary if, for every e∈E, e≤s for some s∈S implies that s is idempotent.Proposition 3.2 (see [1]).
LetS be an inverse semigroup. For s,t∈S, the following are equivalent:(i)
s≤t,(ii)
there  existsf∈E  such  that  s=ft,(iii)
s=ts*s,(iv)
s=ss*t,(v)
s*≤t*.Proposition 3.3.
LetS be an inverse semigroup and e is an idempotent in E. If s∈S such that s≤e, then s is also an idempotent.Proof.
Ifs≤e, then by the previous proposition there exists an idempotent f∈E such that s=ef. Since the semilattice of idempotents is closed under multiplication, we have s∈E.Definition 3.4.
An inverse semigroupS is said to be a 0-E-unitary if, for every nonzero idempotent e, e≤s for some s∈S implies s is idempotent. The components of E* are in the form [s,∞)or(s,∞) for some nonzero element s∈S. By Proposition 3.3, if F is any component of S*=S∖{0}, then F⊆E or F∩E=∅.Lemma 3.5 (see [4]).
IfS is a 0-E-unitary inverse semigroup and s,t∈S are such that s*s=t*t and se=te for some nonzero idempotent e≤s*s, then s=t.Proposition 3.6.
IfS is a 0-E-unitary inverse semigroup with zero, then S is a semilattice with respect to natural order.Proof.
Lets,t∈S. If there is no nonzero u∈S such that u≤s,t, then st=0. So 0 is the infimum of s,t. Now suppose that there exists a nonzero element u such that u≤s,t. By [1], u*u≤s*s and u*u≤t*t. Let f=s*st*t. Then u*u≤f. Setting s1=sf and t1=tf, we have
(3.1)s1*s1=fs*sf=f=ft*tf=t1*t1.
Since
(3.2)s1u*u=sfu*u=su*u=u=tu*u=tfu*u=t1u*u,
by Lemma 3.5 we have s1=t1. So
(3.3)st*t=ss*st*t=sf=s1=t1=tf=ts*s.
Since 0≠u1≤s1,t1 we may apply the above argument to s1,u1,t1 in order to prove that s*tt*=t*ss*, which implies that tt*s=ss*t.
The fact thatu≤s,t implies that su*u=u=tu*u. So
(3.4)t*su*u=t*tu*u=u*u.
Since S is 0-E-unitary, t*s is an idempotent. Also we can prove similarly that ts* is an idempotent. Thus st*t=ts*t=tt*s. Therefore
(3.5)st*t=ts*s=tt*s=ss*t.
We claim that st*t is the infimum of s,t. It is obvious that st*t≤s,t. Since
(3.6)u=su*u=sfu*u=ss*st*tu*u=st*tu*u,
then u≤st*t.Note that ifσ is a representation of an inverse semigroup S in the complex plane (as a Hilbert space), then σ(e)=0or1, for every idempotent element e∈E(S). Such representations are called complex  representations.Now we will fix an inverse semigroupS with 0.Definition 3.7.
A complex representationσ of S on the complex plane is said to be tight if the restriction of σ to E(S) is a tight representation of E(S) in {0,1}.
From the definition one can show that ifs0 is a minimum element of S*=S∖{0}, then χ[s0,∞) is a complex tight representation on S. Also if T is a component of S*, then χT is a complex tight representation on S.
Since every 0-E-unitary inverse semigroup is a semilattice with zero, a representation of S in {0,1} is both a representation of the semilattice S in {0,1} and a complex representation of the inverse semigroup S.Theorem 3.8.
LetS be a 0-E-unitary inverse semigroup and let σ be a representation of S in {0,1}. If σ is tight as a semilattice representation, then it is tight as a complex representation.Proof.
Suppose thatσ is a semilattice tight representation of S in {0,1}. Let F and G be finite subsets of E and H a cover for EF,G. Since E⊆S, then EF,G⊆SF,G. Since H is a cover of EF,G, then there is a cover K of SF,G such that H⊆K. Therefore
(3.7)∑h∈Hσ(h)≤∑k∈Kσ(k),
and hence
(3.8)sgn⁡(∑k∈Kσ(k))≥∏f∈Fσ(f)∏g∈G(1-σ(g)).
Then σ∣E is a tight representation of E in {0,1} and therefore σ is a complex tight representation of S in {0,1}.



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