5 research outputs found
The Largest Subsemilattices of the Endomorphism Monoid of an Independence Algebra
An algebra \A is said to be an independence algebra if it is a matroid
algebra and every map \al:X\to A, defined on a basis of \A, can be
extended to an endomorphism of \A. These algebras are particularly well
behaved generalizations of vector spaces, and hence they naturally appear in
several branches of mathematics such as model theory, group theory, and
semigroup theory.
It is well known that matroid algebras have a well defined notion of
dimension. Let \A be any independence algebra of finite dimension , with
at least two elements. Denote by \End(\A) the monoid of endomorphisms of
\A. We prove that a largest subsemilattice of \End(\A) has either
elements (if the clone of \A does not contain any constant operations) or
elements (if the clone of \A contains constant operations). As
corollaries, we obtain formulas for the size of the largest subsemilattices of:
some variants of the monoid of linear operators of a finite-dimensional vector
space, the monoid of full transformations on a finite set , the monoid of
partial transformations on , the monoid of endomorphisms of a free -set
with a finite set of free generators, among others.
The paper ends with a relatively large number of problems that might attract
attention of experts in linear algebra, ring theory, extremal combinatorics,
group theory, semigroup theory, universal algebraic geometry, and universal
algebra.Comment: To appear in Linear Algebra and its Application
Reconstructing the Topology on Monoids and Polymorphism Clones of the Rationals
We show how to reconstruct the topology on the monoid of endomorphisms of the rational numbers under the strict or reflexive order relation, and the polymorphism clone of the rational numbers under the reflexive relation. In addition we show how automatic homeomorphicity results can be lifted to polymorphism clones generated by monoids
Multicoloured Random Graphs: Constructions and Symmetry
This is a research monograph on constructions of and group actions on
countable homogeneous graphs, concentrating particularly on the simple random
graph and its edge-coloured variants. We study various aspects of the graphs,
but the emphasis is on understanding those groups that are supported by these
graphs together with links with other structures such as lattices, topologies
and filters, rings and algebras, metric spaces, sets and models, Moufang loops
and monoids. The large amount of background material included serves as an
introduction to the theories that are used to produce the new results. The
large number of references should help in making this a resource for anyone
interested in beginning research in this or allied fields.Comment: Index added in v2. This is the first of 3 documents; the other 2 will
appear in physic