1,624 research outputs found
Automorphism groups of countable algebraically closed graphs and endomorphisms of the random graph
We establish links between countable algebraically closed graphs and the
endomorphisms of the countable universal graph . As a consequence we show
that, for any countable graph , there are uncountably many maximal
subgroups of the endomorphism monoid of isomorphic to the automorphism
group of . Further structural information about End is established
including that Aut arises in uncountably many ways as a
Sch\"{u}tzenberger group. Similar results are proved for the countable
universal directed graph and the countable universal bipartite graph.Comment: Minor revision following referee's comments. 27 pages, 3 figure
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
Primitive Groups Synchronize Non-uniform Maps of Extreme Ranks
Let be a set of cardinality , a permutation group on
, and a map which is not a permutation. We say that
synchronizes if the semigroup contains a constant
map.
The first author has conjectured that a primitive group synchronizes any map
whose kernel is non-uniform. Rystsov proved one instance of this conjecture,
namely, degree primitive groups synchronize maps of rank (thus, maps
with kernel type ). We prove some extensions of Rystsov's
result, including this: a primitive group synchronizes every map whose kernel
type is . Incidentally this result provides a new
characterization of imprimitive groups. We also prove that the conjecture above
holds for maps of extreme ranks, that is, ranks 3, 4 and .
These proofs use a graph-theoretic technique due to the second author: a
transformation semigroup fails to contain a constant map if and only if it is
contained in the endomorphism semigroup of a non-null (simple undircted) graph.
The paper finishes with a number of open problems, whose solutions will
certainly require very delicate graph theoretical considerations.Comment: Includes changes suggested by the referee of the Journal of
Combinatorial Theory, Series B - Elsevier. We are very grateful to the
referee for the detailed, helpful and careful repor
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