Incidence Matrices of Directed Graphs of Groups and their up-down Pregroups

The aim of this work is to give a definition of the incidence matrices of the directed graph of groups, construct an up-down pregroup of the incidence matrices of the directed graph of groups and then give an algorithm for the up-down pregroup of the directed graph of groups.


Introduction
n [1] we gave the definition of the incidence Matrices of X-Labeled graphs. In [2], [3] we gave the definition of the directed graph of groups, constructed graph of groups for pregroups directly from the ordered tree of pregroups, and from that directed graph of groups we constructed the up-down pregroups, and then we showed those two pregroups are isomorphic. In [4] Rimlinger gave an example of a pregroup P of finite height; he said "but Jim Shearer and I spent a very long evening with the computer and verified the pregroup axioms". I bear this point in mind. In [2], [3] we have a direct method to obtain examples of pregroups in the form of up-down pregroups from any directed graph of groups, but sometimes those graphs of groups are large, and then will take a long time to find those up-down pregroups. In [1] we defined the incidence matrices of X-labeled graphs. The main aim of this work is to represent the directed graph of finite groups in terms of the incidence matrices of X-labeled graphs, so that by adding certain conditions to allow the incidence matrices of the X-labeled graph to be more confident with the definition of the directed graph of groups; we can then write a computer program to record all elements of the up-down pregroup of that directed graph of groups, as an application of the incidence matrices of X-labeled graph. Therefore, this paper is divided into s i x sections. In section 2, we give the basic concepts of graphs, pregroups and incidence matrices of X-labeled graphs. In section 3, we give the definition of incidence matrices of directed graphs of groups. In section 4, we construct the up-down pregroup of the incidence matrices of the directed graph of groups. In section 5, we define an algorithm on the incidence matrices of the directed graph of groups, so we can then write a computer program for this algorithm.

Basic concepts 2.1 Pregroups
The idea of pregroups goes back to Baer [5] and the definition of pregroup was given independently by Stallings [6] in 1971. The theory of pregroups has been developed by [4], Stallings [6], Hoare [7] and Hoare -Jassim [3] and others. We now return to the original definition of pregroups [6].
Let P be a set with an element 1  P and a mapping of a subset D of P  P into P, denoted by (x, y)  xy. We shall say that xy is defined instead of (x, y)  D. Suppose that there is an involution on P denoted by x  x 1 , such that the following axioms hold: P4: if xy and yz are defined then (xy)z is defined if and only if x(yz) is defined, in which case the two are equal and we will say xyz is defined. P5: For any w, x, y and z in P, if wx, xy and yz are defined, then either wxy or xyz is defined .
Hoare [7] showed that we could prove axiom P3 above by using the following proposition, and axioms P1, P2 and P4.
. It is clear that ~ is an equivalence relation compatible with .

Proposition 2.3.
(i) If x  y or y  x , then x 1 y and y 1 x are defined.
(ii) If xa and a 1 y are defined, then (xa)(a 1 y) is defined if and only if xy is defined, in which case they are equal.  By using axiom P5 above (which will be denoted by P5(i)) Rimlinger [4] proved conditions P5(ii) and P5(iii) of Lemma 2.4 below.
Lemma 2.4 [7]. The following conditions on elements of P are equivalent: P5(i). If wx,xy and yz are defined , then either wxy or xyz is defined . P5(ii). If x 1 a and a 1 y are defined but x 1 y is not , then a < x and a < y.
P5(iii). If x 1 y is defined, then x  y or y  x .  Therefore, we will say P is a pregroup if it satisfies axioms P1, P2, P4, and the conditions of Lemma 2.4, above. The universal group of a pregroup P [13] is denoted by U (P) and has the following presentation  P; x.y  xy whenever xy is defined, for x, y, P . Now if P is a pregroup, then (P, ) is tree -like partial ordering; that is P/~ has a minimum element and, for any x,y and z in P , x  z and y  z we have x  y or y  x . Moreover Rimlinger in [4] defined that for any element x in P, we say that x has finite height n  0, if there exists a maximal totally ordered subset {x 0 , x 1 ,, x n } of P such that 1  x 0  x 1    x n  x . He also showed that the elements of P form an order tree (denoted by O ) whose vertices , [

Incidence Matrices of X -Labeled Graphs
In [1] we gave the definition of the incidence matrices of X -Labeled graphs (where an X-labeled graph is a directed graph with each edge labeled by an element x of the subset X of the group F and X generating the group F), and some definitions and results related to it. Recall that from graph theory the directed graphs  are without loops, because we cannot define the incidence matrices of directed graphs . The incidence matrices of directed graphs  are with n vertices and m edges (i.e. it is and the incidence matrices of the directed graphs do not deal with the labeling of edges, we will put more conditions on the incidence matrices of directed graphs as below to obtain the definition of the incidence matrices of the X-Labeled graphs.
and the X -Labeled graph  has loops with labeling a or b, then choose a mid point on all edges labeled a or b to make all of them two edges labeled aa or bb respectively . Therefore in the rest of this work we will assume that all X -Labeled graphs are without loops.
Now we give the basic definitions and some results on the incidence matrix of X -Labeled graph ) ( X M , as given in [1].     Figure 2. The incidence matrix of the directed graph of groups given in Figure 1 above.

The up-down pregroup of an incidence matrix of a directed graph of finite groups.
In this section we construct the up-down pregroup of the incidence matrix of a directed graph of groups as below; Let ) ( X M be the incidence matrix of a directed graph of groups has the following presentation: