5,651 research outputs found
Racks and blocked braids
In the paper Blocked-braid Groups, submitted to Applied Categorical
Structures, the present authors together with Davide Maglia introduced the
blocked-braid groups BB_n on n strands, and proved that a blocked torsion has
order either 2 or 4. We conjectured that the order was actually 4 but our
methods in that paper, which involved introducing for any group G a braided
monoidal category of tangled relations, were inadequate to demonstrate this
fact. Subsequently Davide Maglia in unpublished work investigated exactly what
part of the structure and properties of a group G are needed to permit the
construction of a braided monoidal category with a tangle algebra and was able
to distinguish blocked two-torsions from the identity.
In this paper we present a simplification of his answer, which turns out to
be related to the notion of rack. We show that if G is a rack then there is a
braided monoidal category TRel_G generalizing that of the above paper. Further
we introduce a variation of the notion of rack which we call irack which yields
a tangle algebra in TRel_G. Iracks are in particular racks but have in addition
to the operations abstracting group conjugation also a unary operation
abstracting group inverse. Using iracks we obtain new invariants for tangles
and blocked braids permitting us to present a proof of Maglia's result that a
blocked double torsion is not the identity.
This work was presented at the Conference in Memory of Aurelio Carboni,
Milan, 24-26 June 2013
Tangled Circuits
The theme of the paper is the use of commutative Frobenius algebras in
braided strict monoidal categories in the study of varieties of circuits and
communicating systems which occur in Computer Science, including circuits in
which the wires are tangled. We indicate also some possible novel geometric
interest in such algebras
Bicategories of spans as cartesian bicategories
Bicategories of spans are characterized as cartesian bicategories in which
every comonad has an Eilenberg-Moore ob ject and every left adjoint arrow is
comonadic
Blocked-braid Groups
We introduce and study a family of groups , called the
blocked-braid groups, which are quotients of Artin's braid groups
, and have the corresponding symmetric groups as
quotients. They are defined by adding a certain class of geometrical
modifications to braids. They arise in the study of commutative Frobenius
algebras and tangle algebras in braided strict monoidal categories. A
fundamental equation true in is Dirac's Belt Trick; that
torsion through is equal to the identity. We show that
is finite for and 3 but infinite for
The compositional construction of Markov processes II
In an earlier paper we introduced a notion of Markov automaton, together with
parallel operations which permit the compositional description of Markov
processes. We illustrated by showing how to describe a system of n dining
philosophers, and we observed that Perron-Frobenius theory yields a proof that
the probability of reaching deadlock tends to one as the number of steps goes
to infinity. In this paper we add sequential operations to the algebra (and the
necessary structure to support them). The extra operations permit the
description of hierarchical systems, and ones with evolving geometry
- …