On Thursday, February 25, 2016 at 3:18:48 PM UTC-6, geneN wrote:
What is known, and what is interesting about the composition of ternary relations? Are there references to ternary relation algebras?
Algebras that are "relativized" have a natural formulation in terms of
ternary operators. For instance, if you relativize a group by removing
the special standing of the identity then the implicit appearance of
the group identity e in product g.h gets made explicit g/e.h and this
ternary operation becomes the fundamental operation -- with axioms
a/b.b = a; a/a.b = b; (a/b.c)/d.e = a/b.(c/d.e) and ... for Abelian
groups ... a/b.c = c/b.a. This structure T is, itself, a bundle that
has at each point T_a a "group" as its fibre given by the "relativized"
group operations x ._a y = x/a.y; e_a = a; x^{-1} = e/x.e. In addition;
one can also define a uniform group dT by taking a formal quotient a\b
modulo the equivalence a\(b/c.d) = (c/b.a)\d with both of these serving
to identity the product operation (a\b)(c\d) in the group dT. T is
called a "torsor" and is the actual structure that's involved in the
geometries underlying gauge theory rather than Lie groups. A Lie
Torsor.
Another example like this occurs if you take the vector space and
remove the special standing of the 0 vector. In this case; you have TWO operations -- one for the product [v,r,w] = (1-r)v + rw; and the
terinary sum v - w + x; the original vector space operations being rv =
[0, r, v] and v + w = v - 0 + w. The first operation, here, is
multi-sorted: v, w are vectors; r lies in the coefficient field and the
result [v, r, w] is a vector.
This time, one of the operations can be defined in terms of the other
if the underlying coefficient field is of size > 2; since v - w + x = [w,1/(1-r),v],r,[w,1/r,x]] for any r other than 0 or 1. If the field is
of size > 3 the sole axioms required are [v, 0, w] = v; [v, 1, w] = w
and [v, rt(1-t), [w, s, x]] = [[v, rt(1-s), w], t, [v, rs(1-t), x]].
For fields of size 3; the operation the brackets introduces is v.w =
[v, -, w] and satisfies the axioms of the commutative version of what's
known as a "quandle" (quandles are algebras used in knot theory) and is
weaker than the axioms for affine geometry over 3-element fields.
Affine geometries over 2-elements fields are not handled by this
approach. But these are already known: they're Boolean algebras.
A similar approach can make the basic operations of a principal bundle; likewise for associated bundles; over into ternary algebras. A hint of
that is already seen with the approach adopted for vector spaces.
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