Cancellative actions

summary:The following problem is considered: when can the action of a cancellative semigroup $S$ on a set be extended to a simply transitive action of the universal group of $S$ on a larger set

A precedence theorem for semigroups

In a finite semigroup, the least element under a precedence order is an idempotent in the kernel

Commutative orders revisited

This article studies commutative orders, that is, commutative semigroups having a semigroup of quotients. In a commutative order S, the square-cancellable elements S(S) constitute a well-behaved separable subsemigroup. Indeed, S(S) is also an order and has a maximum semigroup of quotients R, which is Clifford.We present a new characterisation of commutative orders in terms of semilattice decompositions of S(S) and families of ideals of S. We investigate the role of tensor products in constructing quotients, and show that all semigroups of quotients of S are homomorphic images of the tensor product R ⊗S(S) S. By introducing the notions of generalised order and semigroup of generalised quotients, we show that if S has a semigroup of generalised quotients, then it has a greatest one. For thiswe determine those semilattice congruences on S(S) that are restrictions of congruences on S

Abstract algebra

Designed for an advanced undergraduate- or graduate-level course, Abstract Algebra provides an example-oriented, less heavily symbolic approach to abstract algebra. The text emphasizes specifics such as basic number theory, polynomials, finite fields, as well as linear and multilinear algebra. This classroom-tested, how-to manual takes a more narrative approach than the stiff formalism of many other textbooks, presenting coherent storylines to convey crucial ideas in a student-friendly, accessible manner. An unusual feature of the text is the systematic characterization of objects by universa