Gyenge többségi függvények = Weak near-unanimity operations

Az OTKA pályázatom benyújtása (2008. február 11) óta 8 cikkem jelent nemzetközi folyóiratokban, de ezek közül csak kettő cikk született a kutatási programban megadott témában a beszámolási időszak alatt, ezért a többi 6 megjelent cikket nem tüntettem fel a jelentésben. Ezen cikkeken kívül a kutatási tervnek megfelelően további 3 kézirat van publikálásra benyújtva, illetve egy kézirat született, amely nincsen még benyújtva. A kutatási időszakban összesen hat nemzetközi konferencián adtam elő, ezek közül négyen meghívott előadóként. Az [1] cikkben egy irányított fákból álló speciális gráfosztályra, az úgynevezett speciális triádokra bizonyítjuk a homomorfizmus problémára vonatkozó dichotómia sejtés. A [2] cikkben algoritmust adunk arra, hogy láncok direkt szorzatában az azonos elemszámú ideálok melyikében maximális az elemek magasságának összege. A [3] kéziratban a korlátos szélességű és kevés részhatvánnyal rendelkező algebrákra vonatkozó kényszerkielégíthetőségi problémát megoldó algoritmusokat ötvöztem. A [4] kéziratban bebizonyítottuk a Valeriote sejtést reflexív irányított gráfokra. Az [5] kéziratban a Valeriote sejtés több ekvivalens megfogalmazását adtuk meg. A [6] kéziratban a CSP probléma egy teljesen új redukcióját vezettük be, amely segítségével újabb algebraosztályokra bizonyítható a dichotómia sejtés. | Since the submission of my OTKA grant proposal (2008/08/11) I had eight articles appeared in international journals, but only two of them were on a topic listed in the project proposal. Therefore, I did not mention the other 6 in this report. Beside these articles I have 3 submitted manuscripts and one manuscript not jet submitted. During the three years of this research grant I have given 6 talks on international conferences, four of them were invited plenary talks. In [1] we have proved the constraint satisfaction dichotomy conjecture for a special class of directed trees, for the class of special triads. In [2] we give an algorithm to determine the order ideal of a direct product of chains in which the number of elements equals a fixed integer and the sum of heights of elements is maximal. In [3] I have combined the algorithms solving the constraint satisfaction problem for bounded width algebras and for algebras of few subpowers. In [4] we have proved the Valeriote conjecture for reflexive directed graphs. In [5] we have given equivalent formulations of the Valeriote conjecture. in [6] we have introduced a completely new reduction of CSP problems that allowed the proof of the dichotomy conjecture for various new classes of algebras

The number of clones determined by disjunctions of unary relations

We consider finitary relations (also known as crosses) that are definable via finite disjunctions of unary relations, i.e. subsets, taken from a fixed finite parameter set $\Gamma$. We prove that whenever $\Gamma$ contains at least one non-empty relation distinct from the full carrier set, there is a countably infinite number of polymorphism clones determined by relations that are disjunctively definable from $\Gamma$. Finally, we extend our result to finitely related polymorphism clones and countably infinite sets $\Gamma$.Comment: manuscript to be published in Theory of Computing System

Deciding absorption

We characterize absorption in finite idempotent algebras by means of J\'onsson absorption and cube term blockers. As an application we show that it is decidable whether a given subset is an absorbing subuniverse of an algebra given by the tables of its basic operations

Topological Birkhoff

One of the most fundamental mathematical contributions of Garrett Birkhoff is the HSP theorem, which implies that a finite algebra B satisfies all equations that hold in a finite algebra A of the same signature if and only if B is a homomorphic image of a subalgebra of a finite power of A. On the other hand, if A is infinite, then in general one needs to take an infinite power in order to obtain a representation of B in terms of A, even if B is finite. We show that by considering the natural topology on the functions of A and B in addition to the equations that hold between them, one can do with finite powers even for many interesting infinite algebras A. More precisely, we prove that if A and B are at most countable algebras which are oligomorphic, then the mapping which sends each function from A to the corresponding function in B preserves equations and is continuous if and only if B is a homomorphic image of a subalgebra of a finite power of A. Our result has the following consequences in model theory and in theoretical computer science: two \omega-categorical structures are primitive positive bi-interpretable if and only if their topological polymorphism clones are isomorphic. In particular, the complexity of the constraint satisfaction problem of an \omega-categorical structure only depends on its topological polymorphism clone.Comment: 21 page

MAL’TSEV CONDITIONS, LACK OF ABSORPTION, AND SOLVABILITY

Abstract. We provide a new characterization of several Mal’tsev conditions for locally finite varieties using hereditary term properties. We show a particular example how lack of absorption causes collapse in the Mal’tsev hierarchy, and point out a connection between solvability and lack of absorption. As a consequence, we provide a new and conceptually simple proof of a result of Hobby and McKenzie, saying that locally finite varieties with a Taylor term possess a term which is Mal’tsev on blocks of every solvable congruence in every finite algebra in the variety. 1

Congruence n-permutable varieties

Many experts have been doing research on characterizations of congruence n-permutable varieties in many different ways. In 1973 Hagemann and Mitschke, generalizing Maltsev conditions, provided a simple and nice characterization of congruence n-permutable varieties. We offer our own approach to the characterization of congruence n-permutable varieties, inspired by the Kearnes/Tschantz lemma