3 research outputs found
Markov semigroups, monoids, and groups
A group is Markov if it admits a prefix-closed regular language of unique
representatives with respect to some generating set, and strongly Markov if it
admits such a language of unique minimal-length representatives over every
generating set. This paper considers the natural generalizations of these
concepts to semigroups and monoids. Two distinct potential generalizations to
monoids are shown to be equivalent. Various interesting examples are presented,
including an example of a non-Markov monoid that nevertheless admits a regular
language of unique representatives over any generating set. It is shown that
all finitely generated commutative semigroups are strongly Markov, but that
finitely generated subsemigroups of virtually abelian or polycyclic groups need
not be. Potential connections with word-hyperbolic semigroups are investigated.
A study is made of the interaction of the classes of Markov and strongly Markov
semigroups with direct products, free products, and finite-index subsemigroups
and extensions. Several questions are posed.Comment: 40 pages; 3 figure
Polygraphs: From Rewriting to Higher Categories
Polygraphs are a higher-dimensional generalization of the notion of directed
graph. Based on those as unifying concept, this monograph on polygraphs
revisits the theory of rewriting in the context of strict higher categories,
adopting the abstract point of view offered by homotopical algebra. The first
half explores the theory of polygraphs in low dimensions and its applications
to the computation of the coherence of algebraic structures. It is meant to be
progressive, with little requirements on the background of the reader, apart
from basic category theory, and is illustrated with algorithmic computations on
algebraic structures. The second half introduces and studies the general notion
of n-polygraph, dealing with the homotopy theory of those. It constructs the
folk model structure on the category of strict higher categories and exhibits
polygraphs as cofibrant objects. This allows extending to higher dimensional
structures the coherence results developed in the first half