On EQ-monoids

An EQ-monoid A is a monoid with distinguished subsemilattice L with 1 2 L and such that any a, b 2 A have a largest right equalizer in L. The class of all such monoids equipped with a binary operation that identifies this largest right equalizer is a variety. Examples include Heyting algebras, Cartesian products of monoids with zero, as well as monoids of relations and partial maps on sets. The variety is 0-regular (though not ideal determined and hence congruences do not permute), and we describe the normal subobjects in terms of a global semilattice structure. We give representation theorems for several natural subvarieties in terms of Boolean algebras, Cartesian products and partial maps. The case in which the EQmonoid is assumed to be an inverse semigroup with zero is given particular attention. Finally, we define the derived category associated with a monoid having a distinguished subsemilattice containing the identity (a construction generalising the idea of a monoid category), and show that those monoids for which this derived category has equalizers in the semilattice constitute a variety of EQ-monoids

Systems validation: application to statistical programs

BACKGROUND: In 2003, the United States Food and Drug Administration (FDA) released a guidance document on the scope of "Part 11" enforcement. In this guidance document, the FDA indicates an expectation of a risk-based approach to determining which systems should undergo validation. Since statistical programs manage and manipulate raw data, their implementation should be critically reviewed to determine whether or not they should undergo validation. However, the concepts of validation are not often discussed in biostatistics curriculum. DISCUSSION: This paper summarizes a "Plan, Do, Say" approach to validation that can be incorporated into statistical training so that biostatisticians can understand and implement validation principles in their research. SUMMARY: Validation is a process that requires dedicated attention. The process of validation can be easily understood in the context of the scientific method

Semigroups with if-then-else and halting programs

The "if–then–else" construction is one of the most elementary programming commands, and its abstract laws have been widely studied, starting with McCarthy. Possibly, the most obvious extension of this is to include the operation of composition of programs, which gives a semigroup of functions (total, partial, or possibly general binary relations) that can be recombined using if–then–else. We show that this particular extension admits no finite complete axiomatization and instead focus on the case where composition of functions with predicates is also allowed (and we argue there is good reason to take this approach). In the case of total functions — modeling halting programs — we give a complete axiomatization for the theory in terms of a finite system of equations. We obtain a similar result when an operation of equality test and/or fixed point test is included

Identities in the Algebra of Partial Maps

We consider the identities of a variety of semigroup-related algebras modelling the algebra of partial maps. We show that the identities are intimately related to a weak semigroup deductive system and we show that the equational theory is decidable. We do this by giving a term rewriting system for the variety. We then show that this variety has many subvarieties whose equational theory interprets the full uniform word problem for semigroups and consequently are undecidable. As a corollary it is shown that the equational theory of Clifford semigroups whose natural order is a semilattice is undecidable

Monoids with tests and the algebra of possibly non-halting programs

We study the algebraic theory of computable functions, which can be viewed as arising from possibly non-halting computer programs or algorithms, acting on some state space, equipped with operations of composition, if-then-else and while-do defined in terms of a Boolean algebra of conditions. It has previously been shown that there is no finite axiomatisation of algebras of partial functions under these operations alone, and this holds even if one restricts attention to transformations (representing halting programs) rather than partial functions, and omits while-do from the signature. In the halting case, there is a natural “fix”, which is to allow composition of halting programs with conditions, and then the resulting algebras admit a finite axiomatisation. In the current setting such compositions are not possible, but by extending the notion of if-then-else, we are able to give finite axiomatisations of the resulting algebras of (partial) functions, with while-do in the signature if the state space is assumed finite. The axiomatisations are extended to consider the partial predicate of equality. All algebras considered turn out to be enrichments of the notion of a (one-sided) restriction semigrou

Partial maps with domain and range: extending Schein's representation

The semigroup of all partial maps on a set under the operation of composition admits a number of operations relating to the domain and range of a partial map. Of particular interest are the operations R and L returning the identity on the domain of a map and on the range of a map respectively. Schein [25] gave an axiomatic characterisation of the semigroups with R and L representable as systems of partial maps; the class is a finitely axiomatisable quasivariety closely related to ample semigroups (which were introduced—as type A semigroups—by Fountain, [7]). We provide an account of Schein's result (which until now appears only in Russian) and extend Schein's method to include the binary operations of intersection, of greatest common range restriction, and some unary operations relating to the set of fixed points of a partial map. Unlike the case of semigroups with R and L, a number of the possibilities can be equationally axiomatised

An investigation as to how a computerised multimedia intervention could be of use for practitioners supporting learners with Autism Spectrum Disorder (ASD)

This practice-based action research investigation seeks to make a valuable, original and academic contribution to knowledge in the computing, language, communication and educational fields. The aim was to establish the therapeutic (language and communication skills) and educational (literacy and numeracy skills) use of individual tailored computer games for practitioners supporting learners (end-users) with Autism Spectrum Disorder (ASD). This was achieved through a continuous collaboration of cohorts of computing undergraduate students and academics (the development team) carrying out an assignment for a module designed and successfully led by this PhD student (the researcher). The researcher continually collaborated with practitioners (users – teaching staff and speech and language therapists in schools) of learners with ASD over many years. The researcher developed a Computerised Multimedia Therapeutic/Educational Intervention (CMT/EI) process, which used an iterative holistic Design-For-One approach for developing individual computer games. An action research methodology was adopted using methodological triangulation ‘quantitative’ and ‘qualitative’ data collection methods. This was to ascertain as to how tailor-made computerised multimedia games developed, could be evaluated by the users as being of therapeutic/educational use for their learners (end-users) with ASD. The researcher originated profiles to establish the diversity of each learner’s spectrum of therapeutic/educational autistic needs, preferences, capabilities, likes, dislikes and interests. The researcher orchestrated, collaborated and supervised the whole process from individual profiles completed by the practitioners, through to the profiles used as a baseline, by the development team, and to the designing, developing and evaluating iterative customised personalised computer games. Four hundred and sixty-four learners with ASD (end-users) and forty-nine practitioners (users) from nine educational establishments across the UK participated in this investigation. Two stages were carried out in an initial application procedure (with one school) and prototype procedure (with a further six schools and 2 educational establishments). Stage I - Planning, collection, organisation, Design-For-One approach and development. Stage II - Testing, Evaluation, Monitoring, Reflection and Maintenance. Optimistic ‘quantitative’ and ‘qualitative’ evidence emerged (using content analysis) from the implementation of games in the classroom and the practitioner’s therapeutic and educational evaluation of storyboards and games. The documented positive findings led to a conclusion that personalised games which had been developed over a ten-year period, showed to be of therapeutic/educational use to practitioners and their learners with ASD

Radicals of 0-regular algebras

We consider a generalisation of the Kurosh--Amitsur radical theory for rings (and more generally multi-operator groups) which applies to 0-regular varieties in which all operations preserve 0. We obtain results for subvarieties, quasivarieties and element-wise equationally defined classes. A number of examples of radical and semisimple classes in particular varieties are given, including hoops, loops and similar structures. In the first section, we introduce 0-normal varieties (0-regular varieties in which all operations preserve 0), and show that a key isomorphism theorem holds in a 0-normal variety if it is subtractive, a property more general than congruence permutability. We then define our notion of a radical class in the second section. A number of basic results and characterisations of radical and semisimple classes are then obtained, largely based on the more general categorical framework of L. M\'arki, R. Mlitz and R. Wiegandt as in [13]. We consider the subtractive case separately. In the third section, we obtain results concerning subvarieties and quasivarieties based on the results of the previous section, and also generalise to subtractive varieties some results for multi-operator group radicals defined by simple equational rules. Several examples of radical and semisimple classes are given for a range of fairly natural 0-normal varieties of algebras, most of which are subtractive

Structural assembly in space

A cost algorithm for predicting assembly costs for large space structures is given. Assembly scenarios are summarized which describe the erection, deployment, and fabrication tasks for five large space structures. The major activities that impact total costs for structure assembly from launch through deployment and assembly to scientific instrument installation and checkout are described. Individual cost elements such as assembly fixtures, handrails, or remote minipulators are also presented

Towards developing an industry led educational framework using LEAN approach

The poor performance and inefficient manner in which the construction industry operates has been recognised through a variety of combined government and industry initiatives over the years. A major challenge towards improvement is recognised as lying with education and industry stakeholders actively creating closer and more effective relationships to facilitate a greater mutual understanding. The application of Information Technology (IT) systems can well enhance ‘Lean’ initiatives through improving process flow, reduction of the non-added value activities, better meet customers’ requirements and adding value which will increase the performance of the industry. This paper presents a project that is focused on developing an industry led framework for educational training programmes. The outcomes of two workshops organised with the industry that have resulted in a Continued Professional Development (CPD) training framework comprising of three distinct levels in terms of strategic, operational and technology aspects of that particular key area are discussed. The essence of this work is based on adopting the ‘Lean’ approach and adding value by identifying the IT skills gaps recognised ‘by the industry’ ‘for the industry’ and addressing them in developing training programmes