Sandwich semigroups in diagram categories

This paper concerns a number of diagram categories, namely the partition, planar partition, Brauer, partial Brauer, Motzkin and Temperley–Lieb categories. If [Formula: see text] denotes any of these categories, and if [Formula: see text] is a fixed morphism, then an associative operation [Formula: see text] may be defined on [Formula: see text] by [Formula: see text]. The resulting semigroup [Formula: see text] is called a sandwich semigroup. We conduct a thorough investigation of these sandwich semigroups, with an emphasis on structural and combinatorial properties such as Green’s relations and preorders, regularity, stability, mid-identities, ideal structure, (products of) idempotents, and minimal generation. It turns out that the Brauer category has many remarkable properties not shared by any of the other diagram categories we study. Because of these unique properties, we may completely classify isomorphism classes of sandwich semigroups in the Brauer category, calculate the rank (smallest size of a generating set) of an arbitrary sandwich semigroup, enumerate Green’s classes and idempotents, and calculate ranks (and idempotent ranks, where appropriate) of the regular subsemigroup and its ideals, as well as the idempotent-generated subsemigroup. Several illustrative examples are considered throughout, partly to demonstrate the sometimes-subtle differences between the various diagram categories.

The Erdős–Burgess constant of the multiplicative semigroup of a factor ring of 𝔽q[x]

Let [Formula: see text] be a commutative semigroup endowed with a binary associative operation [Formula: see text]. An element [Formula: see text] of [Formula: see text] is said to be idempotent if [Formula: see text]. The Erdős–Burgess constant of [Formula: see text] is defined as the smallest [Formula: see text] such that any sequence [Formula: see text] of terms from [Formula: see text] and of length [Formula: see text] contains a nonempty subsequence, the sum of whose terms is idempotent. Let [Formula: see text] be a prime power, and let [Formula: see text] be the polynomial ring over the finite field [Formula: see text]. Let [Formula: see text] be a quotient ring of [Formula: see text] modulo any ideal [Formula: see text]. We gave a sharp lower bound of the Erdős–Burgess constant of the multiplicative semigroup of the ring [Formula: see text], in particular, we determined the Erdős–Burgess constant in the case when [Formula: see text] is the power of a prime ideal or a product of pairwise distinct prime ideals in [Formula: see text].

6. Algebra and the arithmetic of remainders

‘Algebra and the arithmetic of remainders’ considers a new type of algebra, which is both an ancient topic and one that has found major contemporary application in Internet cryptography. It begins with an outline of abstract algebra, including groups, rings, and fields. Semigroups and groups are algebras with a single associative operation, while rings and fields are algebras with two operations linked via the distributive law. Lattices are algebras with an ordered structure, while vector spaces and modules are algebras where the members can be multiplied by scalar quantities from other fields or rings. The rules of modular arithmetic (or clock arithmetic) and solving linear congruences are also described.

A UNIFYING LOOK AT SEMIGROUP COMPUTATIONS ON MESHES WITH MULTIPLE BROADCASTING

Given a sequence of m items α1, α2,…, αm from a semigroup S with an associative operation ⊕, the semigroup computation problem involves computing α1 ⊕ α2 ⊕…⊕ αm. We consider the semigroup computation problem involving m (2≤m≤n) items on a mesh with multiple broadcasting of size [Formula: see text]. Our contribution is to present the first lower bound and the first time-optimal algorithm which apply to the entire range of m (2≤m≤n). Specifically, we show that any algorithm that solves the semigroup computation problem must take Ω( log m) time if [Formula: see text] and [Formula: see text] time for [Formula: see text]. We then show that our bound is tight by designing an algorithm whose running time matches the lower bound. These results unify and generalize all semigroup lower bounds and algorithms known to the authors.

Some filters of partitions

§0. Introduction. We started our study of filters of partitions in [15]. We shall here restrict ourselves to the consideration of filters on (ω)ω, the set of all infinite partitions of ω. §1 is an attempt to elucidate the connection between filters on (ω)ω and filters over ω. Given a filter H over ω, we define two filters FH and GH on (ω)ω, and we characterize p-points, rare ultrafilters and Ramsey ultrafilters in terms of properties of the associated filters of partitions.The remainder of the paper is devoted to the study of those filters that can be associated with Hindman's theorem and its extensions. Let us introduce some notation. Suppose * is an associative operation on ω, and let a subset A of ω and an ordinal α with 0 < α ≤ ω be given. We define a collection of subsets of ω by letting iff , where , is an increasing sequence of elements of A for each i < α, and whenever 0 < i < α. Then the Milliken-Taylor theorem (see [17] and [21]) asserts that for every F: [ω/n → m, where n and m are positive integers, there exists A Є [ω]ω such that F is constant on . Hindman's theorem [8] is the special case of this result when n = 1. (We shall conform to usage and simply write FS(A) instead of Glazer (see [6]) has given a proof of Hindman's theorem that uses idempotent ultrafilters. We shall see in §5 that the Milliken-Taylor theorem too can be derived in this fashion.

Embedding a semigroup of transformations

Let X be an arbitrary set and θ a transformation of X. One may use θ to induce an associative operation in Jx, the set of all mappings of X to itself as follows: . We denote the resulting semigroup by {Jx;θ) Magill (1967) introduced this structure and it has been studied by Sullivan and by myself.