Peeter Normak

Monoids over which all flat cyclic right acts are strongly flat

ConclusionsThe table at the end of this paper summarizes the definitive results that have so far been obtained on the classification of monoids by properties of their cyclic right acts, where the properties in question belong to the collection (free, projective, strongly flat, condition (P), flat, weakly flat, torsion free). The presence of question marks in the table indicates that an exact description of the class of monoids in question has not yet been determined (although partial results may be known: for example, Theorem 3.6 of the present paper describes theleft PP monoids over which (weakly) flat cyclic right acts have property (P), [16] and [17] contain further such results.)Among the classes of monoids whose exact descriptions remain unknown if one attempts a classification by properties ofarbitrary rather than cyclic right acts are the following: {S| every rightS-act having (P) is projective} {S| every rightS-act having (P) is strongly flat} {S| every (weakly) flat rightS-act has (P)} {S| every weakly flat rightS-act is flat} {S| every rightS-act is flat}

Flatness properties of monocyclic acts

In a previous paper the authors studied flatness properties of cyclic actsS/ρ (S denotes a monoid, and ρ is a right congruence onS), and determined conditions onS under which all flat or weakly flat acts of this type are actually strongly flat or projective. In the present paper attention is restricted to monocyclic acts (cyclic acts in which ρ is generated by a single pair of elements ofS), and further results on such collapsing of flatness properties are obtained. An observation which is used extensively in this study is the fact that forw andt inS withwt≠t,S/ρ(wt,t) is flat if and only ift is a regular element ofS.

Designing Dippler — A Next-Generation TEL System

This paper discusses the conceptual design of Digital Learning Ecosystems, which, as we argue, are becoming the foundation of next-generation Technology-Enhanced Learning systems (TEL systems). We illustrate our argument by a case study on design and development of a Dippler ecosystem. First, the framework for identifying the generations of TEL systems is described and expectations towards next generation of TEL systems are drawn from the literature review. After that, the dialectics of ongoing mainstream discourse (LMS vs PLE) is analysed and platform for reaching the synthesis is drawn. As we argue, the next-generation TEL systems are better understood if not referred as “learning environments”, but rather as Digital Learning Ecosystems. Finally, process and results of a design-based research on Digital Learning Ecosystem called Dippler is described and discussed.

Delivering QTI Self-tests to Personal Learning Environments Using Wookie Widgets

Although online tests are a quite popular form of assessing learning outcomes in traditional Learning Management Systems, it is quite difficult to deliver the wealth of existing re-usable QTI-compliant tests into Web 2.0 style Personal Learning Environments. This paper addresses both pedagogical scenarios and technical solution for using QTI self-tests in blogs, personal e-portfolios and social media platforms. The empirical part describes the development of the original QTI Assessment Proxy Service and QTI self-test player based on the Wookie widget engine.

Hereditary endomorphism monoids of projective acts

Projective acts whose endomorphism monoids are left or right (semi-) hereditary are characterized. For example, it is shown that for a noncyclic free or projective S-act P, End P is left (semi) hereditary if and only if P ≈ Se1 Π Se2 and e1Se1, e2Se2 are groups.

Morita duality for monoids

In this paper Morita duality for monoids is introduced. Necessary and sufficient conditions for two monoids S and T to be Morita dual are given. Moreover, it is shown that if S and T are Morita dual monoids, then S and U are Morita dual if and only if T and U are Morita equivalent. In addition, every finite monoid having Morita duality is selfdual and even reflexive.

Learning Outcomes for Blog-Based Courses: A Case Study

We specify the learning outcomes achieved through blog-based courses for students who have never experienced this type of course before. We also describe the main problems the students faced during the course, analyse the reasons for these problems and propose some scaffolding possibilities. The study relies on two consecutive presentations of the course “Intelligent Computer Use” in Tallinn University.

On Congruence Compact Monoids

A universal algebra is called congruence compact if every family of congruence classes with the finite intersection property has a non-empty intersection. This paper determines the structure of all right congruence compact monoids S for which Greens relations ℐ and ℋ coincide. The results are thus sufficiently general to describe, in particular, all congruence compact commutative monoids and all right congruence compact Clifford inverse monoids.

Strategic planning of e-learning innovation: Interplay between national and institutional levels

In last few decades E-learning has been established as a recognized form of education. E-learning offers good opportunities for virtual mobility of students-taking courses from other universities. This calls for a need to harmonize strategic planning of e-learning innovation on institutional and inter-institutional (regional, national) levels. The paper will try to point the interplay between those levels. The relation between strategic planning and innovation management, interplay between national and institutional levels of strategic planning is also discussed.

A Model for University Seminars held in Companies

The paper discusses a model for university seminars that take place in ICT companies or IT departments of companies. The presentation examines the main implications in the light of three years of experience in running such seminars.