Topological Group Language Theory
Preliminary Note 3
Boundary of Words
[Theorem]
1
Distance space (X, d)
Gromov product (y|z)x
2
Hyperbolic space X
point sequence {xi
X}
Set of all the {xi
X} S∞(X)
Point sequence of X {yi}
When {yi} is limi→∞(xi|yi) = 0, {yi} is in S∞(X).
[Explanation]
1
Hyperbolicity for general distance space is defined by Gromov product.
Details are below.
2
Distance space is defied by the next.
Distance space that has basic point x0 (X, d)
Arbitrary 3 points of X x, y, z
When (X, d) satisfies the next, it is called δ-hyperbolic.
(x|y)x0
min{(x|z)x0, (y|z)x0}-δ
When distance space (X, d) is δ-hyperbolic for arbitrary base point, X is called δ-hyperbolic.
Here for a certain
, δ-hyperbolic space is abbreviatedly called hyperbolic.
3
The next condition is equivalent with what (X, d) is δ-hyperbolic.
d(x, y) + d(z, w)
max{ d(x, z) + d(y, w), d(x, w) + d(y, z) } +2δ
4
Distance space X
Arc of X α : [0, λ] → X
Arbitrary s, t
[0, λ]
d(α(s), α(t)) = |s-t|
Arc α is called geodesic segment.
Geodesic segment from x to y that is x,y
X is expressed by
.
When arbitrary 2 points of distance space (X, d) are combinable by geodesic segment, X is
called geodesic space.
xyz =
is called geodesic triangle.
[Impression]
1
Word is identified with point sequence {xi
X}.
Language is identified with S∞(X).
New generated word is identified Point sequence of X, {yi} that has condition limi→∞(xi|yi).
The new generated word is also in language, that is to say, S∞(X).
2
From the condition limi→∞(xi|yi), spherical surface is considered as boundary at infinity by
the comparison with Poincaré model.
3
Spherical surface is considered as the unit of language.
Language's distance and warp is also considered under hyperbolic space.
References are below.
[References]
#1 ![]()
Quantum Theorey for Language
#2 ![]()
Distance Theory
#3 ![]()
Warp Theory
The upper papers and the related papers with the themes are seen at ![]()
Sekinan Linguistuic
Field.
To be continued
Tokyo February 12, 2009
