On the (un)decidability of a near-unanimity term

We investigate the near-unanimity problem: given a finite algebra, decide if it has a near-unanimity term of finite arity. We prove that it is undecidable of a finite algebra if it has a partial near-unanimity term on its underlying set excluding two fixed elements. On the other hand, based on Rosenberg’s characterization of maximal clones, we present partial results towards proving the decidability of the general problem

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

Algebraosztályok és klónok = Classes of algebras and clones

Főbb eredményeink a következők. Beláttuk, hogy többségi kifejezésfüggvény létezése eldönthető véges algebrákra. Igazoltuk, hogy bármely nem-triviális idempotens Malcev-feltételt teljesítő véges algebrának van gyenge többségi kifejezésfüggvénye, és 2-uniform kongruenciái felcserélhetőek. Bizonyítottuk, hogy a k-parallelogramma-kifejezéssel rendelkező véges, reziduálisan kicsi algebrák és a véges 2-nilpotens csoportok kifejezésfüggvényeinek klónját véges sok reláció meghatározza. Új dichotómiatételeket kaptunk a kényszer-kielégíthetőségi problémára és az egyenletrendszer-megoldhatósági problémára. Számos új eredményt kaptunk hálók kombinatorikai vonatkozásairól, fraktál- és féligmoduláris hálókról. Struktúratételeket bizonyítottunk az E-tömör lokálisan inverz félcsoportokra, az E-unitér majdnem faktorizálható ortodox félcsoportokra, valamint kiterjesztettük a majdnem faktorizálható inverz félcsoportok elméletét a lokálisan inverz félcsoportok osztályára. Jellemeztünk bizonyos transzformáció-monoidokat, amelyek egyelemű monadikus intervallumot határoznak meg a klónhálóban. Leírtuk a centralizátorklónt véges, egyszerű, idempotens algebrák és bizonyos kongruencia disztributív varietást generáló véges algebrák esetén. Új eredményeket kaptunk 3-változós többségi függvénnyel rendelkező minimális klónokra. Beláttuk, hogy a kompozícióra zárt függvényosztályok hálója kontinuum számosságú a kételmű halmazon, és leírtuk e háló szerkezetét. | Our main results are as follows. We proved that the existence of a near-unanimity term operation is decidable for finite algebras. We showed that if a finite algebra admits a nontrivial idempotent Maltsev condition, then it has a weak near-unanimity term operation, and its 2-uniform congruences permute. We proved that the clone of any finite residually small algebra with a k-parallelogram term operation and any finite 2-nilpotent group is determined by finitely many relations. We obtained new dichotomy theorems for the constraint satisfaction problem and for the solvability problem of systems of equations. We proved a number of theorems on the combinatorial aspects of lattices and on fractal and semimodular lattices. We obtained new structure theorems for E-solid locally inverse semigroups and E-unitary almost factorizable orthodox semigroups. Furthermore we extended the theory of almost factorizable inverse semigroups to the class of locally inverse semigroups. We characterized certain transformation monoids which determine a one-element monoidal interval in the lattice of clones. We described the centralizer clones of finite simple idempotent algebras and of certain algebras in congruence distributive varieties. We obtained new results on the minimal clones containing majority operations. We proved that on the two-element set the lattice of function classes closed under composition has the cardinality of the continuum, and described the structure of this lattice

Datalog and Constraint Satisfaction with Infinite Templates

On finite structures, there is a well-known connection between the expressive power of Datalog, finite variable logics, the existential pebble game, and bounded hypertree duality. We study this connection for infinite structures. This has applications for constraint satisfaction with infinite templates. If the template Gamma is omega-categorical, we present various equivalent characterizations of those Gamma such that the constraint satisfaction problem (CSP) for Gamma can be solved by a Datalog program. We also show that CSP(Gamma) can be solved in polynomial time for arbitrary omega-categorical structures Gamma if the input is restricted to instances of bounded treewidth. Finally, we characterize those omega-categorical templates whose CSP has Datalog width 1, and those whose CSP has strict Datalog width k.Comment: 28 pages. This is an extended long version of a conference paper that appeared at STACS'06. In the third version in the arxiv we have revised the presentation again and added a section that relates our results to formalizations of CSPs using relation algebra

Adjoint functors and tree duality

A family T of digraphs is a complete set of obstructions for a digraph H if for an arbitrary digraph G the existence of a homomorphism from G to H is equivalent to the non-existence of a homomorphism from any member of T to G. A digraph H is said to have tree duality if there exists a complete set of obstructions T consisting of orientations of trees. We show that if H has tree duality, then its arc graph delta H also has tree duality, and we derive a family of tree obstructions for delta H from the obstructions for H. Furthermore we generalise our result to right adjoint functors on categories of relational structures. We show that these functors always preserve tree duality, as well as polynomial CSPs and the existence of near-unanimity functions.Comment: 14 pages, 2 figures; v2: minor revision