ON INCIDENCE ALGEBRAS AND DIRECTED GRAPHS

The incidence algebra I(X,R) of a locally finite poset (X,≤) has been defined and studied by Spiegel and O’Donnell (1997). A poset (V ,≤) has a directed graph (Gv,≤) representing it. Conversely, any directed graph G without any cycle, multiple edges, and loops is represented by a partially ordered set VG . So in this paper, we define an incidence algebra I(G,Z) for (G,≤) over Z, the ring of integers, by I(G,Z)= {fi,f∗ i : V×V → Z}where fi(u,v) denotes the number of directed paths of length i from u to v and f∗ i (u,v) = –fi(u,v). When G is finite of order n, I(G,Z) is isomorphic to a subring ofMn(Z). Principal ideals Iv of (V ,≤) induce the subdigraphs 〈Iv〉 which are the principal ideals v of (Gv,≤). They generate the ideals I( v ,Z) of I(G,Z). These results are extended to the incidence algebra of the digraph representing a locally finite weak poset both bounded and unbounded.


Introduction.
The incidence algebra I(X, R) of a locally finite poset X over a commutative ring R with identity is discussed in much detail by Spiegel and O'Donnell [5].Every poset (V , ≤) has a directed graph (G v , ≤), or (G, ≤) for short, representing it.The directed graph (G, ≤) is free of cycles and multiple arcs.It is natural for (G, ≤) to have an incidence algebra whose properties depend on those of the directed graph.
In this paper, we define and study an incidence algebra I(G, Z) for (G, ≤) over the ring Z of all integers.Section 2 contains the basic results and definitions used in this paper.Section 3 deals with I(G, Z), the incidence algebra of (G, ≤) representing a finite poset (V , ≤) of cardinality n.Ideals of I(G, Z) are given through the principal ideals of (G, ≤) which are the subdigraphs induced by the principal ideals of (V , ≤).An extension of the results given in Section 3 to the digraph representing a locally finite weak poset (V , ≤) bounded or unbounded is the content of Section 4.

Definitions and basic results.
We consider the directed graph (G v , ≤) representing a locally finite partially ordered set (V , ≤).The terminology introduced here is used throughout the paper.
Lemma 2.3 (see [5]).A finite poset (V , ≤) can be labelled Proposition 2.4 (see [4]).The vertices of a finite directed graph G = (V , E) can be labelled v 1 ,v 2 ,...,v n such that Definition 2.6 (see [5]).Let (X, ≤) be a locally finite poset and R a commutative ring with identity.The incidence algebra I(X, R) of X over R is defined by with operations defined by (i) Definition 2.8.An ideal Ᏽ of (G v , ≤) is an induced subdigraph of G v such that all directed paths with their terminal vertex in Ᏽ are in Ᏽ.
If I v is a principal ideal of (V , ≤), I v , the subdigraph induced by I v is the principal ideal of G generated by v in (G v , ≤).Denote I v by Ᏽ v .Notation 2.9.For the remaining part of the paper, the digraphs (G, ≤) and (G ∞ , ≤) represent the finite poset (V , ≤) and the locally finite poset (V ∞ , ≤), respectively.

An incidence algebra for (G, ≤).
In this section, we define an incidence algebra I(G, Z) for the digraph (G, ≤) representing the finite poset (V , ≤).Subalgebras and ideals of I(G, Z) are defined through principal ideals of (G, ≤).Assume that The incidence algebra I(G, Z) of (G, ≤) over the commutative ring Z with identity is defined by Remark 3.2.The function f l is the digraph analogues of χ ∈ I(X, R) [5].The matrix [f l ,(v i ,v j )] is the adjacency matrix of (G, ≤) and of (G, ≤) we associate an incidence algebra, (3.2) Proof.Let Ᏽ v be a proper principal ideal of (G, ≤).Denote the elements of I(G, Z) and I(Ᏽ v , Z) by f and f , respectively.For all f ∈ I(Ᏽ v , Z), there is a unique f ∈ I(G, Z) such that f (u, w) = f (u,w) for all (u, w) ∈ I v × I v ; and f (u, w) = 0 for all (u, w) ∈ I v × I v .
Hence for any (u, w) with similar values for (g f )(u, w) also.Consequently, f g and g f ∈ I(Ᏽ v , Z), and Remark 3.7.The incidence algebra I(G, Z) is isomorphic to a subring of the ring of upper triangular matrices over Z [5].In general, every ideal of M n (Z) has the form M n (S) for some ideal S of Z [3].Proposition 3.8.Every ideal of I(G, Z) has the form I(Ᏽ v , Z) for some principal ideal Ᏽ v of (G, ≤).
Proof.Let S be a proper ideal of the ring I(G, Z).Then S = I(H, Z) for some subdigraph (H, ≤) of (G, ≤).For all f ∈ I(G, Z) and g ∈ S, f g = g f ∈ S.
Consequently, there is an h ∈ S such that f g = h satisfying (f g )(u, v) = h (u, v) = p k (u, v) for some k and for all u, v in V (H).
Then for all (u, v) ∈ V (H)× V (H), the number of paths of length k in H from u to v is the same as that in G.
Hence for any v ∈ V (H), H contains all the directed paths terminating in v. Then H = Ᏽ v for some v.

The incidence algebra I(G ∞ , Z
).An extension of Proposition 2.4 is obtained by the author for locally finite directed graphs [2].This provides an isomorphism between I(G ∞ , Z) and a subring of the ring of upper triangular matrices over Z. Also we have extended Propositions 3.6 and 3.8 to bounded as well as unbounded locally finite weak posets.It is assumed that the poset (V , ≤) is countable.Proposition 4.4 (see [2]).Let G = (V , E) be a locally finite, acyclic digraph where V is countable; satisfying the operations given in Definition 3.1.

Definition 4.6. An infinite matrix
and [f ] is both row and column finite, M ∞ (Z) denotes the ring of row and column finite matrices over Z.The incidence algebra (ii) A characterization of infinite directed graphs for which [f ] is nilpotent is obtained by the author in [1].Proposition 4.8.Let (V , ≤) be a weak, locally finite poset satisfying the following: (i) the set S of its atoms is nonempty and finite; (ii) for each v ∈ V , s ∈ S such that s ≤ v, any chain with v as upperbound is of finite length.Then I(G ∞ , Z) is isomorphic to a subring of the ring of upper triangular matrices over Z.
Proof.The digraph (G ∞ , ≤) is locally finite.By Proposition 4.4, f (v r ,v s ) ≥ 0 for all r ≤ s and for all f ∈ I(G ∞ , Z). Hence every f has a representation as an upper triangular matrix over Z. Definition 4.9.In a bounded weak, locally finite poset (V , ≤) a principal ideal Proposition 4.10.Let (V , ≤) be any weak, locally finite bounded poset and Ᏽ v a principal ideal of (G ∞ , ≤).Then (1) Note 4.11.This follows from Propositions 3.6 and 4.4.
Proposition 4.12 (see [5]).Let (V , ≤) be an unbounded weak poset.Then V contains a subpartially ordered set isomorphic to Z + , Z -, or UCn where Cn denotes a chain of length n.Remark 4.13.When (V , ≤) is unbounded, the principal ideals of (V , ≤) and (G ∞ , ≤) are not well defined.Hence Proposition 4.4 is not true for unbounded posets, in general.
Let (V , ≤) be an unbounded, locally finite, weak poset such that I(G ∞ , Z) is isomorphic to a subring of the ring of upper triangular matrices.Then (i) principal ideals of (G ∞ , ≤) are defined as

Call for Papers
Thinking about nonlinearity in engineering areas, up to the 70s, was focused on intentionally built nonlinear parts in order to improve the operational characteristics of a device or system.Keying, saturation, hysteretic phenomena, and dead zones were added to existing devices increasing their behavior diversity and precision.In this context, an intrinsic nonlinearity was treated just as a linear approximation, around equilibrium points.Inspired on the rediscovering of the richness of nonlinear and chaotic phenomena, engineers started using analytical tools from "Qualitative Theory of Differential Equations," allowing more precise analysis and synthesis, in order to produce new vital products and services.Bifurcation theory, dynamical systems and chaos started to be part of the mandatory set of tools for design engineers.
This proposed special edition of the Mathematical Problems in Engineering aims to provide a picture of the importance of the bifurcation theory, relating it with nonlinear and chaotic dynamics for natural and engineered systems.Ideas of how this dynamics can be captured through precisely tailored real and numerical experiments and understanding by the combination of specific tools that associate dynamical system theory and geometric tools in a very clever, sophisticated, and at the same time simple and unique analytical environment are the subject of this issue, allowing new methods to design high-precision devices and equipment.
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Definition 4 . 1 .
A locally finite poset (V , ≤) is weak if only finitely many chains intersect at every element v in V .Definition 4.2.A directed graph G = (V , E) is locally finite if every vertex is of a finite degree.Definition 4.3.A vertex v of the directed graph (G ∞ , ≤), such that d(v) ≠ 0, is called a source or atom if there is no other vertex u such that u ≤ v.

Definition 4 . 5 .
its sources is nonempty and finite; (ii) for each v ∈ V and s ∈ S such that → sv ∈ E, any maximal directed path terminating in v is of finite length.Let (V , ≤) be a locally finite weak poset and (G ∞ , ≤) the digraph representing (V , ≤).The incidence algebra I(G ∞ , Z) of (G ∞ , ≤) over the ring Z of integers is given by

First
Round of ReviewsMarch 1, 2009 and only if G has no cycles.