40 research outputs found
Pattern avoidance classes and subpermutations
Pattern avoidance classes of permutations that cannot be expressed as unions
of proper subclasses can be described as the set of subpermutations of a single
bijection. In the case that this bijection is a permutation of the natural
numbers a structure theorem is given. The structure theorem shows that the
class is almost closed under direct sums or has a rational generating function.Comment: 18 pages, 4 figures (all in-line
Decidability of well quasi-order and atomicity for equivalence relations under embedding orderings
We consider the posets of equivalence relations on finite sets under the
standard embedding ordering and under the consecutive embedding ordering. In
the latter case, the relations are also assumed to have an underlying linear
order, which governs consecutive embeddings. For each poset we ask the well
quasi-order and atomicity decidability questions: Given finitely many
equivalence relations , is the downward closed set
Av consisting of all equivalence relations which do not
contain any of : (a) well-quasi-ordered, meaning that it
contains no infinite antichains? and (b) atomic, meaning that it is not a union
of two proper downward closed subsets, or, equivalently, that it satisfies the
joint embedding property
Heights of one- and two-sided congruence lattices of semigroups
The height of a poset is the supremum of the cardinalities of chains in
. The exact formula for the height of the subgroup lattice of the symmetric
group is known, as is an accurate asymptotic formula for the
height of the subsemigroup lattice of the full transformation monoid
. Motivated by the related question of determining the heights
of the lattices of left- and right congruences of , we develop a
general method for computing the heights of lattices of both one- and two-sided
congruences for semigroups. We apply this theory to obtain exact height
formulae for several monoids of transformations, matrices and partitions,
including: the full transformation monoid , the partial
transformation monoid , the symmetric inverse monoid
, the monoid of order-preserving transformations
, the full matrix monoid , the partition
monoid , the Brauer monoid and the
Temperley-Lieb monoid
The Bergman property for semigroups
In this article, we study the Bergman property for semigroups and the associated notions of cofinality and strong cofinality. A large part of the paper is devoted to determining when the Bergman property, and the values of the cofinality and strong cofinality, can be passed from semigroups to subsemigroups and vice versa. Numerous examples, including many important semigroups from the literature, are given throughout the paper. For example, it is shown that the semigroup of all mappings on an infinite set has the Bergman property but that its finitary power semigroup does not; the symmetric inverse semigroup on an infinite set and its finitary power semigroup have the Bergman property; the Baer-Levi semigroup does not have the Bergman property.PostprintPeer reviewe
Generators and relations for subsemigroups via boundaries in Cayley graphs
Given a finitely generated semigroup S and subsemigroup T of S we define the notion of the boundary of T in S which, intuitively, describes the position of T inside the left and right Cayley graphs of S. We prove that if S is finitely generated and T has a finite boundary in S then T is finitely generated. We also prove that if S is finitely presented and T has a finite boundary in S then T is finitely presented. Several corollaries and examples are given.PostprintPeer reviewe
Gendering Latin American Independence: Database and Image Bank
It is well known that if two algebraic structures A and B are residually finite then so is their direct product. Here we discuss the converse of this statement. It is of course true if A and B contain idempotents, which covers the case of groups, rings, etc. We prove that the converse also holds for semigroups even though they need not have idempotents. We also exhibit three examples which show that the converse does not hold in general.PostprintPeer reviewe
Finite presentability of Bruck-Reilly extensions of Clifford monoids
Let M be a Clifford monoid and let theta be an endomorphism of M. We prove that if the Bruck-Reilly extension BR(M,theta) is finitely presented then M is finitely generated. This allows us to derive necessary and sufficient conditions for Bruck-Reilly extensions of Clifford monoids to be finitely presented.</p