ternary relation composition

[geneN]

What is known, and what is interesting about the composition of ternary
relations? Are there references to ternary relation algebras?

[rockbr...@gmail.com]

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.

[geneN]

On Thursday, February 25, 2016 at 4:18:48 PM UTC-5, geneN wrote:

> What is known, and what is interesting about the composition of ternary
> relations? Are there references to ternary relation algebras?

Image understanding uses a betweenness relation [a,b,c] to denote the
object b is between objects a and c, and there is a notion of
transitivity (whenever composition) that is appropriate to that field.
 The notion of composition however is unsatisfying as the analog of
binary relation composition for a variety of reasons.  I have
investigated some conditions under which a relation notion of
composition results in the operation being associative, but my
assumptions are somewhat ad hoc and not satisfying.  For example, the
identity relation, whose matrix has 1's on the main diagonal and 0's
elsewhere, isn't the identity element in this algebra.

I think you can find my notes on this at research gate.com, On Ternary
Relation Composition, by Eugene M. Norris

[Kat]

On Thursday, February 25, 2016 at 1:18:48 PM UTC-8, geneN wrote:

> What is known, and what is interesting about the composition of ternary
> relations? Are there references to ternary relation algebras?

Ternary relations are very interesting and absolutely needed in
Abstract algebra. Where binary relations (a,b) are not sufficient as in
a relationship with an x & y axis. Ternary relations (a,b,c) are needed
to resolve and explain relationships with x, y and z axis, where z is a
plane.

[rockbr...@gmail.com]

On Tuesday, August 23, 2016 at 11:23:24 AM UTC-5, rockbr...@gmail.com
wrote:

> Algebras that are "relativized" have a natural formulation in terms of
> lgebras that are "relativized" have a natural formulation in terms of

> ternary operators. [e.g. Vector Space relativized to Affine Geometry] 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.

[Affine geometry]
I posted a detailed development of this in smr in the 1990's under
"Mark's Elements". This, here,

http://orion.math.iastate.edu/jdhsmith/math/FA2PGvC.pdf

is the first reference I've ever found that uses the same ternary
algebra; but from a cursory reading it appears they don't find the
unifying formulation that handles most or all the fields that I cited
(which, by the way, was uncovered via a partially automated
"theory-generation" process). The operation I alluded to AB = [A, -1,
B] for the 3-element field is generic to all characteristic 3 fields --
it's simply the midpoint. The properties AA = A; AB = BA; A(AB) = B;
A(BC) = (AB)(AC) continue to hold in that setting. Other fields with
odd or 0 characteristic also have the midpoint operator; only those
with characteristic 22 don't. The paper separately treats the cases of
even, odd and zero characteristic fields.