Functorial Calculus in Monoidal Bicategories

The definition and calculus of extraordinary natural transformations is extended to a context internal to any autonomous monoidal bicategory. The original calculus is recaptured from the geometry of the monoidal bicategory V-Mod whose objects are categories enriched in a cocomplete symmetric monoidal category V and whose morphisms are modules.     

幺半群上的广义内射S-系(英文)

本文考虑拟内射、伪内射、核内射以及支内射S-系的性质,重点讨论这些广义内射S-系的上的线性方程组的性质.例如,A是核内射S-系当且仅当A的任意收缩核B上的任意A-相容线性方程组在B上是可解的;A是支内射S-系当且仅当同构于A的任一分支的S-系B上的任意A-相容线性方程组在B上是可解的.进而讨论了伪内射、核内射以及支内射S-系的融合余积的性质.最后给出了一个充分条件,基于此条件核内射和支内射S-系是等价的.     

Principal Weak Flatness and Regularity of Diagonal Acts

For a monoid S, the set S × S equipped with the componentwise right S-action is called the diagonal act of S and is denoted by D(S). A monoid S is a left PP (left PSF) monoid if every principal left ideal of S is projective (strongly flat). We shall call a monoid S left P(P) if all principal left ideals of S satisfy condition (P). We shall call a monoid S weakly left P(P) monoid if the equalities as = bs, xb = yb in S imply the existence of r ∈ S such that xar = yar, rs = s. In this article, we prove that a monoid S is left PSF if and only if S is (weakly) left P(P) and D(S) is principally weakly flat. We provide examples showing that the implications left PSF ? left P(P) ? weakly left P(P) are strict. Finally, we investigate regularity of diagonal acts D(S), and we prove that for a right PP monoid S the diagonal act D(S) is regular if and only if every finite product of regular acts is regular. Furthermore, we prove that for a full transformation monoid S = 𝒯 _X , D(S) is regular.     

Eulerian* -polynomial identities on matrix algebras

In this paper we considerably strengthen a result of Ribenboim on noetherian generalised power series rings. While Ribenboim proves his result under the restrictive assumption that the monoid occuring in the definition of the geralised power series ring in cancellative we prove a corresponding result for arbitrary ordered monoids.     

On Some Finitary Conditions Arising from the Axiomatisability of Certain Classes of Monoid Acts

This article considers those monoids S satisfying one or both of the finitary properties (R) and (r), focussing for the most part on inverse monoids. These properties arise from questions of axiomatisability of classes of S-acts, and appear to be of interest in their own right. If S is weakly right noetherian (WRN), that is, S has the ascending chain condition on right ideals, then certainly (r) holds. Other than this, we show that (R), (r), and (WRN) are independent. Our most detailed results are for Clifford monoids, in which case we completely characterise those S with trivial structure homomorphisms satisfying (R) or (r).     

A Presentation of the Singular Part of the Symmetric Inverse Monoid#

We give a semigroup presentation of the singular part of the symmetric inverse monoid on a finite set. Along the way, we derive a monoid presentation of the monoid of all order-preserving injective partial transformations on a finite chain, which differs from the presentation discovered by Fernandes.     

Ideal Structure of the Kauffman and Related Monoids

The generators of the Temperley-Lieb algebra generate a monoid with an appealing geometric representation. It has been much studied, notably by Louis Kauffman. Borisavljevi?, Do?en, and Petri? gave a complete proof of its abstract presentation by generators and relations, and suggested the name “Kauffman monoid”. We bring the theory of semigroups to the study of a certain finite homomorphic image of the Kauffman monoid. We show the homomorphic image (the Jones monoid) to be a combinatorial and regular *-semigroup with linearly ordered ideals. The Kauffman monoid is explicitly described in terms of the Jones monoid and a purely combinatorial numerical function. We use this to describe the ideal structure of the Kauffman monoid and two other of its homomorphic images.     

On Von Neumann Regular Rings of Skew Generalized Power Series

In this paper we introduce a construction called the skew generalized power series ring R[[S, ω]] with coefficients in a ring R and exponents in a strictly ordered monoid S which extends Ribenboim's construction of generalized power series rings. In the case when S is totally ordered or commutative aperiodic, and ω(s) is constant on idempotents for some s ∈ S?{1}, we give sufficient and necessary conditions on R and S such that the ring R[[S, ω]] is von Neumann regular, and we show that the von Neumann regularity of the ring R[[S, ω]] is equivalent to its semisimplicity. We also give a characterization of the strong regularity of the ring R[[S, ω]].     

Delta Sets of Numerical Monoids Using Nonminimal Sets of Generators

Several recent articles have studied the structure of the delta set of a numerical monoid. We continue this work with the assumption that the generating set S chosen for the numerical monoid M is not necessarily minimal. We show that for certain choices of S, the resulting delta set can be made (in terms of cardinality) arbitrarily large or small. We close with a close analysis of the case where M =?n ₁, n ₂, in ₁ + jn ₂?for non-negative i and j.     

Override and update

Override and update are natural constructions for combining partial functions, which arise in various program specification contexts. We use an unexpected connection with combinatorial geometry to provide a complete finite system of equational axioms for the first order theory of the override and update constructions on partial functions, resolving the main unsolved problem in the area.