22 research outputs found
The conjugacy problem for automorphism groups of countable homogeneous structures
We consider the conjugacy problem for the automorphism groups of a number of
countable homogeneous structures. In each case we find the precise complexity
of the conjugacy relation in the sense of Borel reducibility
On Two-Path Convexity in Multipartite Tournaments
Abstract In the context of two-path convexity, we study the rank, Helly number, Radon number, Caratheodory number, and hull number for multipartite tournaments. We show the maximum Caratheodory number of a multipartite tournament is 3. We then derive tight upper bounds for rank in both general multipartite tournaments and clone-free multipartite tournaments. We show that these same tight upper bounds hold for the Helly number, Radon number, and hull number. We classify all clone-free multipartite tournaments of maximum Helly number, Radon number, hull number, and rank. Finally we determine all convexly independent sets of clone-free multipartite tournaments of maximum rank
Some relational structures with polynomial growth and their associated algebras II: Finite generation
The profile of a relational structure is the function which
counts for every integer the number, possibly infinite, of
substructures of induced on the -element subsets, isomorphic
substructures being identified. If takes only finite values, this
is the Hilbert function of a graded algebra associated with , the age
algebra , introduced by P.~J.~Cameron.
In a previous paper, we studied the relationship between the properties of a
relational structure and those of their algebra, particularly when the
relational structure admits a finite monomorphic decomposition. This
setting still encompasses well-studied graded commutative algebras like
invariant rings of finite permutation groups, or the rings of quasi-symmetric
polynomials.
In this paper, we investigate how far the well know algebraic properties of
those rings extend to age algebras. The main result is a combinatorial
characterization of when the age algebra is finitely generated. In the special
case of tournaments, we show that the age algebra is finitely generated if and
only if the profile is bounded. We explore the Cohen-Macaulay property in the
special case of invariants of permutation groupoids. Finally, we exhibit
sufficient conditions on the relational structure that make naturally the age
algebra into a Hopf algebra.Comment: 27 pages; submitte
Recommended from our members
The life and work of Major Percy Alexander MacMahon
This thesis describes the life and work of the mathematician Major Percy Alexander MacMahon (1854 - 1929). His early life as a soldier in the Royal Artillery and events which led to him embarking on a career in mathematical research and teaching are dealt with in the first two chapters. Succeeding chapters explain the work in invariant theory and partition theory which brought him to the attention of the British mathematical community and eventually resulted in a Fellowship of the Royal Society, the presidency of the London Mathematical Society, and the award of three prestigious mathematical medals and four honorary doctorates. The development and importance of his recreational mathematical work is traced and discussed. MacMahon's career in the Civil Service as Deputy Warden of the Standards at the Board of Trade is also described. Throughout the thesis, his involvement with the British Association for the Advancement of Science and other scientific organisations is highlighted. The thesis also examines possible reasons why MacMahon's work, held in very high regard at the time, did not lead to the lasting fame accorded to some of his contemporaries. Details of his personal and social life are included to give a picture of MacMahon as a real person working hard to succeed in a difficult context