• The function coverz constructs a finite part of the universal covering of a zero relation algebra, large enough to decide about finite or infinite representation type.

• A zero relation algebra z is given in terms of a quiver and relations by a triple
  
  [n,a,r]
  
  Here:
  • n is the number of vertices of the quiver of z. The vertices are then identified with the numbers 1, ..., n.

  • a is a list
    
    a = [a₁, ..., a_u]
    
    containing the arrows of the quiver of z, each arrow a_i being a triple of integers
    
    a_i = [s, t, h]
    
    where s is the start vertex of the arrow a_i, t its terminal vertex and h a label assigned to it. Different arrows may have the same label.

  • r is a list
    
    r = [r₁, ..., r_v]
    
    describing the relations imposed in the following way:
    Each list entry r_i is a list
    
    r_i = [l₁, ... ,l_w]
    
    of labels indicating that any composition p₁ ... p_w of arrows with the arrow p_j having label l_j is zero within z.

• The universal covering of a zero relation algebra is given by a (usually infinite) tree with relations. The function returns a finite part of this, in the format of a bound poset algebra (see posetalgebra).

• The finite part of the covering is constructed in the following way:
  
  The algorithm starts with a spanning tree of the original quiver. To this spanning tree, in a width first fashion, further vertices are added, unless one of the following tree situations is encountered:
  
  A vertex j will receive no further adjacent vertices if
  1. There is no nonzero path from the vertex j to the initial spanning tree.
  2. The vertex is found to lie on an infinite chain sequence.
  3. The algebra is found not to be finite dimensional.

• Information on the constructed part of the covering is passed via the optional arguments of the function:
  • The optional second argument, if given, will be assigned a list of integers
    
    c = [ c[1], ..., c[N] ]
    
    where N is the number of vertices of the part of the covering constructed indicating that the vertex i of the covering lies over the vertex c[i] of the original algebra.

  • The optional third argument, if given, will be assigned a list of integers
    
    v=[ v[1], ..., v[N] ]
    
    that for each vertex i contains its state with respect to the algorithm sketched above in encoded form. The explicit meaning of the entries is given by the following table:

    v[i]	Status of the vertex i
    -1	The vertex belongs to the original spanning tree. In this case, all of its neighbours in the universal covering are contained in the part constructed.
    -2	All of the neighbours of the vertex in the universal covering are contained in the part constructed.
    2	There is no nonzero walk from the vertex in question to the initial spanning tree. There might be neighbours in the universal covering not contained in the constructed part.
    3	The vertex lies on an infinite chain. There might be neighbours in the universal covering not contained in the constructed part.
    4	Processing this vertex, the original algebra has been found not to be finite dimensional. Processing has been aborted.

  • The optional forth argument, if given, will be assigned the maximum of the event codes assigned to the optional third argument v. In particular, a value of 3 means, that the original algebra is not representation finite.
    A value of 2 means, that the original algebra is representation finite if and only if the constructed part of the covering is. (In fact, the covering is large enough to contain for every vertex of the original algebra a covering vertex, for which the poset of minimal chain sequences (cf rstartsets) can completely be computed.)

SEE ALSO: posetalgebra, mchrepfin, rstartsets