A REMARK ON THE EXTENSION OF THE CONCEPT OF INCIDENCE ALGEBRAS TO NONLOCALLY

IJMMS 2004:40, 2145–2147 PII. S016117120430325X http://ijmms.hindawi.com

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A REMARK ON THE EXTENSION OF THE CONCEPT OF INCIDENCE ALGEBRAS TO NONLOCALLY

FINITE PARTIALLY ORDERED SETS

BONIFACE I. EKE Received 4 March 2003

An incidence algebra of a nonlocally finite partially ordered setQis a very rare concept, perhaps nonexistent. In this note, we will attempt to construct such an algebra.

2000 Mathematics Subject Classification: 06A11.

1. Introduction. LetPbe a partially ordered set (poset) andKa field of character- istic 0. The functionsf :P×P→K, such thatxy impliesf (x,y)=0, are called the incidence functions ofP overK. The set of such functions is denoted byϑ(K,P).

Pis called locally finite if for everyx,y∈Pthe interval[x,y]= {t∈P|x≤t≤y}is finite. WhenPis locally finite,ϑ(K,P)becomes aK-algebra under a multiplication(∗) defined by convolution:

f∗g(x,y)=

x≤t≤y

f (x,t)g(t,y), (1.1)

and the algebraϑ(K,P)is called the incidence algebra ofP overK[1,2].

IfP is not locally finite, the expression (1.1) may not make sense. So, one does not often hear of an incidence algebra of a nonlocally finite poset. Our purpose in this note is to show that ifQis any nonlocally finite poset andP is a locally finite poset, we can form a nonlocally finite posetQS(P)for which an incidence algebraϑ(K,QS(P))can be constructed.

Moreover, the posetsQandPare both embeddable inQS(P), while the setϑ(K,Q) and the algebraϑ(K,P) are both embeddable inϑ(K,QS(P)), and if|P| ≤ |Q|, then

|QS(P)| = |Q|. Besides, for the fixed posetsPandQ, the incidence algebraϑ(K,QS(P)) is unique up to isomorphism. All these are established inSection 2.

In Section 3, we isolate the auxiliary locally finite posetP and try to deal directly with Q. However, because of the problem still posed by (1.1), we can only construct a sequence of what are called truncated incidence algebras for the nonlocally finite posetQ. For this purpose, we will need an additional hypothesis thatQis well ordered.

2. The construction ofQS(P)andϑ(K,QS(P)). We will assume throughout thatP is a locally finite poset,Qa nonlocally finite poset, andKis a field of characteristic 0.

LetQS(P)be the Cartesian productP×Q. We will denote the order relation inPby≤⁽¹⁾ and the order relation inQby≤⁽²⁾. Then we define an order relation≤inQS(P)by

2146 BONIFACE I. EKE

(x,r )≤(y,s)if and only ifx≤⁽¹⁾yandr ≤⁽²⁾s. It is clear that, with the relation≤, QS(P)is a partially ordered set. However,QS(P)is not locally finite.

We will define addition and scalar multiplication onϑ(K,QS(P))as in [1]. We now need to define the convolution multiplication shown in (1.1) onϑ(K,QS(P))so that it will make sense.

For a fixedr ∈Q, denoteP× {r}byPr. If(x,r )and (y,s)are any two elements of QS(P), then (x,r ) ∈P_r, while (y,s)∈P_s. Moreover, P_r and P_s are locally finite subposets ofQS(P). Denote(x,r ) byuand (y,s)byv, and letT = {t∈P|x≤⁽¹⁾ t≤⁽¹⁾y}. Then the setT (u,v)=(T× {r})∪(T× {s})⊆QS(P)is finite. LetJ(u,v)= [u,v]∩T (u,v). We define the operation(∗)onϑ(K,QS(P))by the following: for all elementsuandvinQS(P)and for allf ,ginϑ(K,QS(P)),

f∗g(u,v)=

z∈J(u,v)

f (u,z)g(z,v). (2.1)

Clearly, (2.1) is now well defined. The associativity follows from [1, Proposition 4.1].

Consequently, with (2.1),ϑ(K,QS(P))is an incidence algebra ofQS(P)overK.

P is isomorphic toPr for eachr∈Q. Similarly, for eachy∈P,Qis isomorphic to Q_y= {y}×Q. Hence bothPandQare embeddable inQS(P). Moreover, the correspon- denceµ_r:f f_r, wheref_r is defined byf_r(x_r,y_r)=f (x,y), is an isomorphism of ϑ(K,P)ontoϑ(K,Pr). Consequently,ϑ(K,P)is embeddable inϑ(K,QS(P)). By a simi- lar device, we find thatϑ(K,Q)is also embeddable inϑ(K,QS(P)). For the uniqueness ofϑ(K,QS(P)), we will prove the following.

Proposition2.1. IfPandQare any posets such thatPis isomorphic toPandQ is isomorphic toQ, thenϑ(K,QS(P))is isomorphic toϑ(K,QS(P)).

Proof. Letσ:P→Pbe an isomorphism whileθ:Q→Qis an isomorphism. Define η:QS(P)→QS(P)byη(x,r )=(σ (x),θ(r )). Ifη(x,r )=η(y,s), then(σ (x),θ(r ))= (σ (y),θ(s)). By the definition of the order relation inQS(P), we must haveσ (x)= σ (y) and θ(r )=θ(s). Consequently,x =y and r =s. Hence,(x,r )=(y,s). This shows thatηis injective. Clearly, alsoηis surjective. Therefore,ηis an isomorphism.

For eachu∈QS(P), denoteη(u)byu. Now defineβ:ϑ(K,QS(P))→ϑ(K,QS(P)) byβ(f )=f, wherefis defined byf(u,v)=f (u,v)for allu,v∈QS(P). One can directly check thatβis also an isomorphism. Hence, the proposition holds.

We observe that for the locally finite posetP, one could have chosen any nonempty finite subset ofQitself. We will call the algebraϑ(K,QS(P))the incidence algebra of Qrelative toP.

3. Truncated incidence algebras. Our interest now is to see what we can achieve by isolating the locally finite posetP and dealing directly withQ. However, (1.1) still poses a problem. Nevertheless, following the motivation received fromSection 2, what we need is to try to use a finite number of elements of the interval[r ,s]at a time, for any two elementsrandsof the nonlocally finite posetQ. Then arises the question of how to choose the finite number of elements from[r ,s]. The formula for choosing such elements is outlined below for the case whereQis well ordered. What makes it possible

A REMARK ON THE EXTENSION OF THE CONCEPT OF INCIDENCE... 2147 is the property of a well-ordered set whereby not only does every nonempty subset of such a set have a first element, but also such a first element is unique [3, Theorems 64 and 65, page 76]. First, we show the existence of a well-ordered nonlocally finite posetQ.

Example3.1. LetQ= {1/n|n∈N} ∪ {O}, whereNis the set of natural numbers.

Qis a poset subject to the usual relation “≥” (greater than or equal to). Clearly, also Qis well ordered by “≥”. However, for anya∈Q,a≠0, the interval[0,a]is infinite.

Hence,Qis not locally finite.

Now letW be any well-ordered poset and let r ≤s∈W. SetW0=[r ,s]. LetW1= W0− {r}. Then, if W1≠∅, W1 has a unique first element t1. Let W2=w1− {t1}. If W2≠∅, thenW2 has a unique first elementt2. In general,Wi=W_i−1− {t_i−1}, where t_i−1=first element ofW_i−1, andt0≡r.

For any fixed natural numbern, letT_n(r ,s)= {r ,t1,...,t_n,s}. Let

J_n(r ,s)=





[r ,s] if[r ,s]is finite,

T_n(r ,s) otherwise. (3.1)

Then define the convolution multiplication∗onϑ(K,W )by the following: for allr ,s∈ Wand for allf ,g∈ϑ(K,W ),

f∗g(r ,s)=

t∈Jn(r ,s)

f (r ,t)g(t,s). (3.2)

Subject to (3.2),ϑ(K,W )is an incidence algebra. We denote this incidence algebra by ϑ_n(K,W ), andϑ_n(K,W )is called atruncated incidence algebraofWoverK.

It is clear thatTn(r ,s)⊆T_n+1(r ,s)for alln∈N. We will call the incidence algebra ϑ_n+1(K,W )arefinement of the incidence algebraϑn(K,W ). The sequence{ϑn(K,W )} of incidence algebras is finite if and only ifWis locally finite.

We now observe that a well-ordered nonlocally finite posetQis associated with an infinite sequence of truncated incidence algebras, where each is a nontrivial refinement of the one before it. Unifying these algebras to form one incidence algebra ofQoverK remains an open problem.

References

[1] M. Aigner,Combinatorial Theory, Grundlehren der mathematischen Wissenschaften, vol.

234, Springer-Verlag, New York, 1979.

[2] R. P. Stanley,Structure of incidence algebras and their automorphism groups, Bull. Amer.

Math. Soc.76(1970), 1236–1239.

[3] P. Suppes,Axiomatic Set Theory, Dover Publications, New York, 1972.

Boniface I. Eke: Department of Mathematics, Morgan State University, Baltimore, MD 21251, USA

E-mail address:[email protected]