13th International Conference on Membrane Computing - MTA Sztaki

A. Obtu̷lowicz
2−ramification
•
diagram,
i.e. graph
homomorphism
Fig. 2.
.
category representing
hierarchical system
second level
colim D 0
•
D 0
•
• •
• colim D 2
first level
colim D 1 • • colim D 3
•
•
• • • • • •
D 1 D 2 D3
•
• •
0−level
•
• • • •
the fat arrows are colimiting injections,
i.e. the elements of colimiting cocons,
respectively
3 Inspiring Examples
Following the idea of drawing hypercubes a from [25] we show the examples of
sketch-like multigraphical membrane systems which approach this idea in some
formal way.
For natural numbers n > 0 and i ∈ {1, 2, 3} we define sketch-like multigraphical
membrane systems S i n in the following way:
– the underlying tree T i n of S i n is such that
• the set V (T i n) of vertices is the set of all strings (sequences) of length
not greater than n of digits in D 1 = {0, 1} for i = 1, in D 2 = {0, 1, 2, 3}
for i = 2, and in D 3 = {0, 1, 2, 3, 4, 5, 6, 7} for i = 3,
• the set E(T i n ) of edges of Ti n is such that E(Ti n ) = {(Γ j, Γ ) | {Γ j, Γ } ⊂
V (T i n) and j ∈ D i } with source and target functions being the projections
on the first and the second component, respectively, where Γ j is
the string obtained by juxtaposition a new digit j on the right end of Γ ,
– the family ( G Γ | Γ ∈ V (T i n )) of directed graphs of Sn i is such that for every
non-elementary vertex Γ ∈ V (T i n ) the Γ -diagram Dg(Γ ) is determined in
the following way:
• for i = 1 the diagram Dg(Γ ) is a graph consisting of a single edge
Γ 0 → Γ 1,
a for a notion of a hypercube see [19], [6], [23]
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