James Renshaw

Monoids for which condition (P) acts are projective

S-acts satisfying conditions (P) are projective is given. We also give a new characterisation of those monoids for which all cyclic right S-acts satisfying condition (P) are projective, similar in nature to recent work by Kilp [6]. In addition we give a sufficient condition for all right S-acts that satisfy condition (P) to be strongly flat and show that the indecomposable acts that satisfy condition (P) are the locally cyclic acts.

Software tools for computer aided learning in mathematics

This article aims to describe a number of software tools designed for the preparation of computer‐aided learning materials (CAL) in mathematics. The CALM Project, based in the Mathematics Department of the Heriot‐Watt University in Edinburgh, is one of the projects currently receiving funding from the Computer Board of the United Kingdom as part of the Computers in Teaching Initiative within British universities. This paper will explain the features of software design favoured by the CALM Project team and it will deal not only with CAL in mathematics but also with CAL in general. The CALM Project provides a practical example of software development in the programming language of Pascal within an educational environment.

Some problems of mathematical CAL

This paper describes briefly some problems of mathematical CAL and suggests ways to combat them. Within the context of the CALM Project for Computer Aided Learning in Mathematics we highlight three main areas of difficulty—mathematical display, input and evaluation. These problems are illustrated using examples from software developed at the Heriot-Watt University in Edinburgh and at Southampton University; the examples are taken from both calculus and algebra.

On Adequate Transversals

We consider adequate transversals of abundant semigroups and prove that, in a particular case, there is a natural embedding of an inverse transversal within a certain regular subsemigroup. We also introduce the concepts of simplistic, perfect, and quasi-adequate transversal and provide a number of interesting connections between these.

Adequate Transversals of Quasi-Adequate Semigroups

The concept of an adequate transversal of an abundant semigroup was introduced by El-Qallali in [8] whilst in [7], he and Fountain initiated the study of quasi-adequate semigroups as natural generalisations of orthodox semigroups. In this work we provide a structure theorem for adequate transversals of certain types of quasi-adequate semigroup and from this deduce Saitos classic result on the structure of inverse transversals of orthodox semigroups. We also apply it to left ample adequate transversals of left adequate semigroups and provide a structure for these based on semidirect products of adequate semigroups by left regular bands.

Perfect amalgamation bases

In [6], Howie considered the following problem: If [U; Si] is an amalgam of semigroups and if Ti are subsemigroups of Si such that [U; Ti]is an amalgam, is it true that ?U* Ti, the free product of the amalgam [U; Ti], is embeddable in ?U* Si the free product of the amalgam [U; Si]? He proved, among other things, that if U and Ti are unitary in Si, then the free products are embeddable. We extend these results here using the homological techniques introduced in [4, 7]and culminate in describing those amalgamation bases which always have this property.

Extension and amalgamation in rings

It is proved that if a ring R has the extension property in containing ring S i , then the amalgam [ R; S i ,] is strongly embeddable. Using a result of P. M. Cohn, a necessary and sufficient condition for a ring to be an amalgamation base is then given. It is also shown that R is a level subring of a ring S if and only if S/R is flat. From this, a classical result of P. M. Cohn on flat amalgams is proved.

SOME OBSERVATIONS ON LEFT ABSOLUTELY FLAT MONOIDS

If S is any monoid a (unital) left S-set B is called fiat if the functor -| (from right S-sets into sets) preserves all embeddings of right S-sets, and weakly flat if this functor preserves embeddings of right ideals into S. S is called (weakly) left absolutely flat if all left S-sets are (weakly) flat. For a more complete discussion of these concepts consult [2] and the references cited therein. If S is a semigroup then OL(a,b) (resp. t3R(a,b)) will denote the smallest left (resp. right) congruence on S containing (a,b). Our first two observations concern weak left absolute flatness, and will make use of the following result:

Stability and amalgamation in monoids

Renshaw [(a) Proc. Lond. Math. Soc. 1986, 52 (3), 119–141; (b) Ph.D. Thesis, University of St. Andrews, 1985] proved a collection of results on flatness, extension, and amalgamation for semigroups, and the concept of stability for S-acts was introduced in (a) and shown to be connected to amalgamation. We extend these results here by developing the “homological” structure of amalgamated free products of monoids and provide a new condition for amalgamation of semigroups which is shown to be weaker than those given in these references. We also extend one of the main results in (a).

On Free Products and Amalgams of Pomonoids

The study of amalgamation in the category of partially ordered monoids was initiated by Fakhuruddin in the 1980s. In 1986 he proved that, in the category of commutative pomonoids, every absolutely flat commutative pomonoid is a weak amalgmation base and every commutative pogroup is a strong amalgamation base. Some twenty years later, Bulman-Fleming and Sohail in 2011 extended this work to what they referred to as pomonoid amalgams. In particular, they proved that pogroups are poamalgmation bases in the category of pomonoids. Sohail, also in 2011, proved that absolutely poflat commutative pomonoids are poamalgmation bases in the category of commutative pomonoids. In the present article, we extend the work on pomonoid amalgams by generalizing the work of Renshaw on amalgams of monoids and extension properties of acts over monoids.