Principle of Divisibility for Discrete Monoids

Abstract

A concrete monoid over a category C is a subset of
the endomorphisms of an object of C containing the
identity and closed under composition To contrast
an abstract monoid is just a one object category.
There is a natural notion of division between
concrete monoids distinct from the usual division
of abstract monoids This concrete division is
identied via two examples and then dened giving
rise to a bicategory of concrete monoids over C
whose arrows are concrete divisions The Poincare
classes of the arrows of this bicategory are found to
have a simple and appealing characterization
allowing us to dene a category of concrete monoids
over C .
These denitions are illustrated with examples from
the theories of semigroups, matrices, vines and
automata With the aid of these denitions, we make
functorial the well known constructions of the
action monoid of an automaton, and the
endomorphism monoid of an object of a category.