László Zádori

TAYLOR TERMS, CONSTRAINT SATISFACTION AND THE COMPLEXITY OF POLYNOMIAL EQUATIONS OVER FINITE ALGEBRAS

We study the algorithmic complexity of determining whether a system of polynomial equations over a finite algebra admits a solution. We characterize, within various families of algebras, which of them give rise to an NP-complete problem and which yield a problem solvable in polynomial time. In particular, we prove a dichotomy result which encompasses the cases of lattices, rings, modules, quasigroups and also generalizes a result of Goldmann and Russell for groups [15].

The Complexity of the Extendibility Problem for Finite Posets

For a finite poset P let EXT(P) denote the following decision problem. Given a finite poset Q and a partial map f from Q to P, decide whether f extends to a monotone total map from Q to P. It is easy to see that EXT(P) is in the complexity class NP. In [SIAM J. Comput., 28 (1998), pp. 57-104], Feder and Vardi define the classes of width 1 and of bounded strict width constraint satisfaction problems for finite relational structures. Both classes belong to the broader class of bounded width problems in P. We prove that for any finite poset P, if EXT(P) has bounded strict width, then it has width 1. In other words, if a poset admits a near unanimity operation, it also admits a totally symmetric idempotent operation of any arity. In [Fund. Inform., 28 (1996), pp. 165-182], Pratt and Tiuryn proved that SAT(P), a polynomial-time equivalent of EXT(P) is NP-complete if P is a crown. We generalize Pratt and Tiuryns result on crowns by proving that EXT(P), is NP-complete for any finite poset P which admits no nontrivial idempotent Malcev condition.

Natural duality via a finite set of relations

We present a duality theorem. We give a necessary and sufficient condition for any set of algebraic relations to entail the set of all algebraic relations in Davey and Werners sense. The main result of the paper states that for a finite algebra a finite set of algebraic relations yields a duality if and only if the set of all algebraic relations can be obtained from it by using four types of relational constructs. Finally, we prove that a finite algebra admits a natural duality if and only if the algebra has a near unanimity term operation, provided that the algebra possesses certain 2 k -ary term operations for some k . This is a generalisation of a theorem of Davey, Heindorf and McKenzie.

Algebraic properties and dismantlability of finite posets

Abstract We show that every finite connected poset which admits certain operations such as Gumm or Jonsson operations, or a near unanimity function is dismantlable. This result is used to prove that a finite poset admits Gumm operations if and only if it admits a near unanimity function. Finite connected posets satisfying these equivalent conditions are characterized by the property that their idempotent subalgebras are dismantlable. As a consequence of these results we obtain that the problem of determining if a finite poset admits a near unanimity function is decidable.

Relational Sets and Categorical Equivalence of Algebras

We study relation varieties, i.e. classes of relational sets (resets) of the same type that are closed under the formation of products and retracts. The notions of an irreducible reset and a representation of a reset are defined similarly to the ones for partially ordered sets. We give a characterization of finite irreducible resets. We show that every finite reset has a representation by minimal resets which are certain distinguished irreducible retracts. It turns out that a representation by minimal resets is a smallest one in some sense among all representations of a reset. We prove that non-isomorphic finite irreducible resets generate different relation varieties. We characterize categorical equivalence of algebras via product and retract of certain resets associated with the algebras. In the finite case the characterization involves minimal resets. Examples are given to demonstrate how the general theorems work for particular algebras and resets.

Monotone Jónsson operations and near unanimity functions

We define nonextendible colored posets and zigzags of a poset. These notions are related to the earlier notions of gaps, holes, obstructions and zigzags considered by Duffus, Nevermann, Rival, Tardos and Wille. We establish some properties of zigzags. By using these properties we give a proof of the well known conjecture that states that any finite bounded poset which admits Jńsson operations, also admits a near unanimity function. We also provide an infinite poset that shows that we cannot drop the finiteness in this conjecture.

GENERATION OF FINITE PARTITION LATTICES

This chapter presents ways to construct cl-generating sets and generating sets of minimum cardinality for the partition lattices P n that are congruence lattices. Strietz has shown that the minimum number of generators of P n ( n ≧ 4) equals 4, and, as a union of ordered chains, a 4-element generating set is of the form 1+1+1+1 or 1+1+2. Strietz also gave a generating set of type 1+1+1+1 for all n ≧ 4 and one of type 1+1+2 for all n ≧ 10. These results imply that the minimum number r of cl-generators of P n is at most 4. The chapter presents a proof that r = 3 provided n ≧ 3, n ≠ 4. Using the 3-element cl-generating set of P n , a new 4-element generating sets of the type 1+1+1+1 for n ≧ 4 and ones of the type 1+1+2 for n ≧ 7 are constructed. Therefore, the existence of a generating set of the form 1+1+2 remains open for n = 5,6.

A polynomial-time algorithm for near-unanimity graphs

We present a simple polynomial-time algorithm that recognises reflexive, symmetric graphs admitting a near-unanimity operation. Several other characterisations of these graphs are also presented.

Reflexive digraphs with near unanimity polymorphisms

Abstract In this paper, we prove that if a finite reflexive digraph admits Gumm operations, then it also admits a near unanimity operation. This is a generalization of similar results obtained earlier for posets and symmetric reflexive digraphs by the second author and his collaborators. In the special case of reflexive digraphs, our new result confirms a conjecture of Valeriote that states that any finite relational structure of finite signature that admits Gumm operations also admits an edge operation. We also prove that every finite reflexive digraph that admits a near unanimity operation admits totally symmetric idempotent operations of all arities. Finally, the aforementioned results yield a polynomial-time algorithm to decide whether a finite reflexive digraph admits a near unanimity operation.

OMITTING TYPES, BOUNDED WIDTH AND THE ABILITY TO COUNT

We say that a finite algebra 𝔸 = 〈A; F〉 has the ability to count if there are subalgebras C of 𝔸3 and Z of 𝔸 such that the structure 〈A; C, Z〉 has the ability to count in the sense of Feder and Vardi. We show that for a core relational structure A the following conditions are equivalent: (i) the variety generated by the algebra 𝔸 associated to A contains an algebra with the ability to count; (ii) 𝔸2 has the ability to count; (iii) the variety generated by 𝔸 admits the unary or affine type. As a consequence, for CSPs of finite signature, the bounded width conjectures stated in Feder–Vardi [10], Larose–Zadori [17] and Bulatov [5] are identical.