LU konference
Ph.D. supervisor: Associate Professor Dr. math. Jānis Cīrulis, University of Latvia.
Reviewers:
Leading Researcher Dr. math. Pēteris Daugulis (Daugavpils University, Latvia);
Professor Dr. math. Gregor Dolinar (University of Ljubljana, Slovenia);
Emeritus Professor Dr. math. Kalle Kaarli (University of Tartu, Estonia).
Abstract
This thesis is concerned with several partial orders on certain classes of Rickart rings (possibly one-sided ones) satisfying that or other strongness condition – a condition which ensures that, without the need for an involution, the ring resembles a Rickart *-ring in certain aspects. This approach allows for the transfer of knowlegde from the field of Rickart *-rings to a wider class of
rings not necessarily having involution, showing that the results obtained for Rickart *-rings can often be at least partly preserved.
In this spirit, we introduce a version of the strong right star order to so-called right-strong Rickart rings, obtaining a relatively orthocomplemented poset like for a Rickart *-ring.
Parts of the thesis focus on investigating conditions for the existence of meets and joins under several partial orders on Rickart rings satisfying a strongness condition. For example, we obtain a series of equivalent conditions under which two elements of a so-called strong Rickart ring have the
join under the star order.
By a unified approach that involves weak BCK-algebras, we obtain, for most of the studied
partial orders, conditions under which the respective poset is a meet semilattice.
Another focus of the thesis is on strong semilattice decompositions of certain reducts of reduced Rickart rings. Among other things, it is shown that, given a reduced Rickart ring (R, +, ·, 1) with the unary operation ◦ mapping every element to its minimal idempotent duplicator, the algebra
(R, ·,◦, 1) is a strong semilattice of right-cancellative D-semigroups, and that this strong semilattice representation is essentially unique.
The doctoral thesis and its summary are available at the UL Library, Raiņa Boulevard 19.
We kindly ask interested participants to register for the session by May 21, 2025, by emailing: janis.bajars@lu.lv
 Academic Center, House of Science, Faculty of Science and Technology, Room 105, Jelgavas Street 3, as well as online via Microsoft Teams, Insa Krēmere will defend her doctoral thesis “Orderings and Structure of Strong Rickart Rings” in an open meeting of the UL Promotion Council in the field of Mathematical Sciences. The degree sought is Doctor of Science (Ph.D.) in Natural Sciences, field: Mathematics; subfield: Algebra and Mathematical Logic.){kind=link}