2006/2007
SEMESTR LETNI 2006/2007
12. 06. 2007, K. Matczak, Quasirozmaitości przemiennnych i symetrycznych grupoidów modowych, c.d.
05. 06. 2007, K. Matczak, Quasirozmaitości przemiennych i symetrycznych grupoidów modowych.
Abstrakt: Opisanie pewnych quasirozmaitości CBM-grupoidów oraz SIE-grupoidów, i ich baz quasirównosciowych. Porównanie struktury krat, jakie tworzą (kilka podobieństw i wiele różnic). Opis niektórych relatywnie podprosto nierozkładalnych grupoidów w opisanych quasirozmaitościach.
29. 05. 2007, M. Stronkowski, Modules in universal algebra
Abstract: I will present some, older and newer results about representation of algebras as (sub)reducts of algebras (polynomially) equvalent to modules.
22. 05. 2007, M. Stronkowski, On some infinite sentences
Abstract: Well known theorem of Shafaat states that any SP-closed class of algebras in a given langauge may be axiomatized by a class of infinite sentences called implications. We will formulate a theorem of that sort for P_S-closed classes of models. Our result is connected with Lyndon theorem about axiomatization of elementary P_S-closed classes of models, but our reasoning is rather in a Shafaat's style. Some limitations of our result will be indicated.
15. 05. 2007, A. Pilitowska, Semiprojection modes
Abstract: Semiprojection modes form one of subclasses of ternary differential modes non-embeddable into semimodules. We give some examples of such non-Szendrei modes, and show that each subvariety of semiprojection modes is relatively based by a set of identities in three variables (not necessarily by only one such identity).
08. 05. 2007, T. Brengos, On covariety and quasi-covariety lattices
Abstract: Basic properties of covariety and quasi-covariety lattices are shown. An attempt to charaterize all covariety lattices is made. A description of the covariety lattice for the identity functor is given. A description of the quasi-covariety lattice of subquasi-covarieties of a covariety generated by a cycle Id-coalgebra is presented.
24. 04. 2007, A. Romanowska, Mody różniczkowe
Abstrakt: M. Stronkowski scharakteryzował mody zanurzalne w półmoduły nad przemiennymi półpierścieniami, i podał pierwsze przykłady modów nie posiadających tej własności. D. Stanovsky skonstruował prosty 3-elementowy przykład ternarnej algebry tego rodzaju. Pokażemy, że algebra Stanovsky'ego należy do klasy modów ternarnych uogólniającej klasę tzw. grupoidów rózniczkowych, omówimy pewne własności tej klasy, i wskażemy dalsze przykłady modów, które nie zanurzają się w półmoduły nad przemiennymi półpierścieniami.
17. 04. 2007, A. Kravchenko (Rosyjska Akademia Nauk, Nowosybirsk) The lattice of quasivarieties of differential groupoids
Abstract: Explicit description of the lattice of varieties of differential groupoids is well known. The lattice L of quasivarieties is Q-universal, which means that no convenient explicit descriptions of this lattice exist. However, we may consider lattices of subquasivarieties of varieties and study their complexity. In the talk, we distinguish some simple and complicated fragments of L and discuss the methods of the proofs.
03. 04. 2007, P. Dehornoy (Uniwersytet w Caen, Francja), 1. From sets to braids via self-distributivity
Abstract: The recent work of H. Woodin significantly renewed Set Theory by restoring its global unity and making it more understandable. For the first time, there exists a reasonable hope of solving the Continuum Problem, and it is very interesting to discuss what this means in view of the classical undecidability results by Godeland Cohen. At the very least, the results by Woodin show that the Continuum Problem is not just a meaningless scholastic question.
27. 03. 2007, M. Ploscica (Słowacka Akademia Nauk), Remarks on Congruence Lattice Problem
Abstract: Investigation of congruences and congruence lattices is one of the main topics in universal algebra and lattice theory. Despite the intensive research, there still are many difficult and challenging problems. The Congruence Lattice Problem is the following question: Is every distributive algebraic lattice isomorphic to the congruence lattice of a lattice? After more than 60 years of effort, the problem has been solved negatively by F. Wehrung in 2005. We will present the main idea of his proof, discuss possible generalizations and open problems. Using Wehrung's technique we prove new results concerning congruence lattices of majority algebras and congruence-permutable algebras.
20. 03. 2007, A. Radzikowska, Representation theorems for some classes of lattice-based algebras
Abstract: In this talk we present representation theorems for some classes of algebras based on not necessarily distributive lattices. Our methodology is an extension of Urquhart representation theorem for lattices and Allwein and Dunn developments on Kripke semantics for linear logics. The representation algebras provide a Kripke-style semantics for the respective classes of multi-valued logics.
13. 03. 2007, A. Radzikowska, An algebraic characterization of multi-valued modal operators
Abstract: In this talk we propose a multi-valued generalisation of modal operators. Some classes of residuated lattices are taken as a basic algebraic structures. Formal properties of these operators are presented. In particular, we show how properties of binary multi-valued relations can be characterised by means of these operators.
06. 03. 2007, Przegląd czasopism, ciąg dalszy
27. 02. 2007, Sprawy organizacyjne, sprawozdania z konferencji, przegląd czasopism
SEMESTR ZIMOWY 2006/2007
16. 01. 2007, Przegląd czasopism, ciąg dalszy
09. 01. 2007, Przegląd czasopism
19. 12. 2006, T. Brengos, Kraty ko-rozmaitości automatów, ciąg dalszy
12. 12. 2006, T. Brengos, Kraty ko-rozmaitości automatów
Abstract: Deterministic automata can be defined as coalgebras of a bounded functor. Therefore Birkhoff Coalgebraic Theorem holds. This yields a characterization of all co-varieties. The aim of this talk is to characterize the co-variety lattice of deterministic automata, show its basic properties and generalize the idea to coalgebras of a functor that preserves arbitrary intersections.
05. 12. 2006, M. Semenova (Rosyjska Akademia Nauk, Nowosibirsk), Lattices of algebraic subsets
Abstract: We are interested in the question which lattices embed into lattices of algebraic subsets of (complete) algebraic lattices. The latter is konown to be isomorphic to quasivariety lattices. We describe some classes of lattices which embed into those.
28. 11. 2006, A. Pilitowska, Kompleksowe algebry grafowe, ciąg dalszy
21. 11. 2006, A. Pilitowska, Kompleksowe algebry grafowe (Complex graph algebras)
Abstract: The complex (global) algebra of (A,F) is the set of all non-ampty subsets of the set A with complex operations of operations from F. G. Graetzer and H. Lakser proved that for a variety V, the complex variety CmV generated by complex algebras of algebras in V, satisfies precisely those identities resulting through identification of variables from the linear identities true in V. In this talk we apply this result to determine all the complex varieties of the variety of entropic graph algebras. We also show identities satisfied in the variety generated by complex algebras of subalgebras of graph algebras.
14. 11. 2006, M. Stronkowski, Finite basis theorems without Mal'cev conditions, ciąg dalszy
07. 11. 2006, M. Stronkowski, Finite basis theorems without Mal'cev conditions
Abstract: The celebrated Baker's theorem says that if a given finite algebra A of a finite similarity type generates congruence distributive variety V(A), then V(A) has finite equational basis. Many proofs of this theorem appeared during last thirty years (Makkai's, Taylor's, Burris's, Jonsson's for example). But in all these (old) proofs Jonsson terms were used. Five years ago a new proof was published by Baker and Wang. It obviously relies on the distributivity of congruence lattices, but does not use Jonsson terms. We will present this proof and discuss a possibility of proving more general finite basis theorems without using of Mal'cev conditions. Our new results are obtained together with Anvar Nurakunov.
31. 10. 2006, G. Bińczak, J. Kaleta, O równoważności entropicznych quasigrup z quasijedynką i przemiennych grup abelowych z inwolucją
Abstrakt: W referacie będzie opisana równoważność pomiędzy entropicznymi quasigrupami z elemantem 1 takim, że x1=x i 1(1x)=x a algebrami (G,+,0,*), gdzie (G,+) jest grupą abelową oraz 0*=0, (x+y)*=x*+y*, x**=x.
24. 10. 2006, A. Kravchenko, Complexity of Q-lattices, kontynuacja
17. 10. 2006, A. Kravchenko (Rosyjska Akademia Nauk, Novosibirsk), Complexity of Q-lattices
Abstract: We consider several approaches to the formalisation of the notion of a complicated Q-lattice and turn to more detailed study of the notion of a Q-universal lattice which was introduced by Sapir in 1985. We compare known sufficient conditions for Q-universality and present a list of Q-universal quasivarieties. We also consider the following question: How to prove that some particular quasivariety is not Q-universal?
10. 10. 2006, A. Romanowska, Abstrakcyjne algebry barycentryczne
Abstrakt: Real convex sets can be presented algebraically with binary operations given by weighted means, the weights taken from the open unit interval in the real numbers. The class of convex setsis a quasivariety (a class defined by certain implications) and generates the variety (a class defined by identities) of so-called barycentric algebras. Both these classes have a well developed theory. However in the specification of convex sets and barycentric algebras, the open unit interval itself has not hitherto been axiomatized. This talk will discuss the problem of axiomatization, and propose one possible solution. We extend the open unit interval of operations to the closed one, and consider barycentric algebras as two-sorted algebras, one sort corresponding to the set of elements of a traditional barycentric algebra, and the second corresponding to a certain algebra of fuzzy logic, a so-called LP-algebra. The LP-algebras yield an algebraic description of the closed unit interval and suggest interesting extension of the class of barycentric algebras. The new structures encompass two-sorted counterparts of barycentric algebras over any ordered field, and also include other algebras providing links to such notions as Boolean affine spaces, $B$-sets of Bergman and Stokes, and ``if-then-else'' algebras.
03. 10. 2006, Sprawy organizacyjne, sprawozdania z konferencji i wizyt naukowych