Finite Idempotent Algebra Research Articles

Universal Algebraic Methods for Constraint Satisfaction Problems

After substantial progress over the last 15 years, the "algebraic CSP-dichotomy conjecture" reduces to the following: every local constraint satisfaction problem (CSP) associated with a finite idempotent algebra is tractable if and only if the algebra has a Taylor term operation. Despite the tremendous achievements in this area (including recently announce proofs of the general conjecture), there remain examples of small algebras with just a single binary operation whose CSP resists direct classification as either tractable or NP-complete using known methods. In this paper we present some new methods for approaching such problems, with particular focus on those techniques that help us attack the class of finite algebras known as "commutative idempotent binars" (CIBs). We demonstrate the utility of these methods by using them to prove that every CIB of cardinality at most 4 yields a tractable CSP.

Existence of cube terms in finite algebras

We study the problem of whether a given finite algebra with finitely many basic operations contains a cube term; we give both structural and algorithmic results. We show that if such an algebra has a cube term then it has a cube term of dimension at most N, where the number N depends on the arities of basic operations of the algebra and the size of the basic set. For finite idempotent algebras we give a tight bound on N that, in the special case of algebras with more than \(\left( {\begin{array}{c}|A|\\ 2\end{array}}\right) \) basic operations, improves an earlier result of K. Kearnes and Á. Szendrei. On the algorithmic side, we show that deciding the existence of cube terms is in P for idempotent algebras and in EXPTIME in general. Since an algebra contains a k-ary near unanimity operation if and only if it contains a k-dimensional cube term and generates a congruence distributive variety, our algorithm also lets us decide whether a given finite algebra has a near unanimity operation.

DECIDING SOME MALTSEV CONDITIONS IN FINITE IDEMPOTENT ALGEBRAS

AbstractIn this paper we investigate the computational complexity of deciding if the variety generated by a given finite idempotent algebra satisfies a special type of Maltsev condition that can be specified using a certain kind of finite labelled path. This class of Maltsev conditions includes several well known conditions, such as congruence permutability and having a sequence of n Jónsson terms, for some given n. We show that for such “path defined” Maltsev conditions, the decision problem is polynomial-time solvable.

On the number of clonoids

A clonoid is a set of finitary functions from a set A to a set B that is closed under taking minors. Hence clonoids are generalizations of clones. By a classical result of Post, there are only countably many clones on a 2-element set. In contrast to that, we present continuum many clonoids for $$A = B = \{0,1\}$$. More generally, for any finite set A and any 2-element algebra $${\mathbf {B}}$$, we give the cardinality of the set of clonoids from A to $${\mathbf {B}}$$ that are closed under the operations of $${\mathbf {B}}$$. Further, for any finite set A and finite idempotent algebra $${\mathbf {B}}$$ without a cube term (with $$|A|,|B|\ge 2$$) there are continuum many clonoids from A to $${\mathbf {B}}$$ that are closed under the operations of $${\mathbf {B}}$$; if $${\mathbf {B}}$$ has a cube term there are countably many such clonoids.

Deciding the Existence of Minority Terms

AbstractThis paper investigates the computational complexity of deciding if a given finite idempotent algebra has a ternary term operation $m$ that satisfies the minority equations $m(y,x,x)\approx m(x,y,x)\approx m(x,x,y)\approx y$. We show that a common polynomial-time approach to testing for this type of condition will not work in this case and that this decision problem lies in the class NP.

Taylor term does not imply any nontrivial linear one-equality Maltsev condition

It is known that any finite idempotent algebra that satisfies a nontrivial Maltsev condition must satisfy the linear one-equality Maltsev condition (a variant of the term discovered by Siggers and refined by Kearnes, Markovic, and McKenzie): $$\begin{aligned} t(r,a,r,e)\approx t(a,r,e,a). \end{aligned}$$ We show that if we drop the finiteness assumption, the k-ary weak near unanimity equations imply only trivial linear one-equality Maltsev conditions for every $$k\ge 3$$ . From this it follows that there is no nontrivial linear one-equality condition that would hold in all idempotent algebras having Taylor terms. Miroslav Olsak has recently shown that there is a weakest nontrivial strong Maltsev condition for idempotent algebras. Olsak has found several such (mutually equivalent) conditions consisting of two or more equations. Our result shows that Olsak’s equation systems cannot be compressed into just one equation.

On finite Taylor algebras

We establish a new hereditary characterization of finite idempotent Taylor algebras. This generalizes the algebraic constructions which figured in the recent successful characterization of the correctness of the bounded width algorithm for constraint satisfaction problems by finite idempotent algebras generating congruence meet-semidistributive varieties. We introduce the cyclic reduct of a finite Taylor algebra and prove a collapse of Mal’cev conditions.

Deciding absorption

We characterize absorption in finite idempotent algebras by means of Jónsson 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.

Idempotent n -permutable varieties

One of the important classes of varieties identified in tame congruence theory is the class of varieties which are n-permutable for some n. In this paper, we prove two results: (1) for every n > 1 , there is a polynomial-time algorithm that, given a finite idempotent algebra A in a finite language, determines whether the variety generated by A is n-permutable and (2) a variety is n-permutable for some n if and only if it interprets an idempotent variety that is not interpretable in the variety of distributive lattices.

Finitely Related Clones and Algebras with Cube Terms

Aichinger et al. (2011) have proved that every finite algebra with a cube-term (equivalently, with a parallelogram-term; equivalently, having few subpowers) is finitely related. Thus finite algebras with cube terms are inherently finitely related—every expansion of the algebra by adding more operations is finitely related. In this paper, we show that conversely, if A is a finite idempotent algebra and every idempotent expansion of A is finitely related, then A has a cube-term. We present further characterizations of the class of finite idempotent algebras having cube-terms, one of which yields, for idempotent algebras with finitely many basic operations and a fixed finite universe A, a polynomial-time algorithm for determining if the algebra has a cube-term. We also determine the maximal non-finitely related idempotent clones over A. The number of these clones is finite.

A Subalgebra Intersection Property for Congruence Distributive Varieties

Abstract. We prove that if a finite algebra Agenerates a congruence distributive variety, then the subalgebras of the powers of Asatisfy a certain kind of intersection property that fails for finite idempotent algebras that locally exhibit affine or unary behaviour. We demonstrate a connection between this property and the constraint satisfaction problem.

Existence theorems for weakly symmetric operations

A k-ary weak near-unanimity operation (or k-WNU) on A is an operation that satisfies the equations w(x, . . . x) ≈ x and w(y, x, . . . , x) ≈ w(x, y, . . . , x) ≈ · · · ≈ w(x, x, . . . , x, y) . If an algebra A has a k-NU (or a k-WNU) term operation, we say that A satisfies NU(k) (or WNU(k), respectively). Likewise, a variety is said to satisfy NU(k) (or WNU(k), respectively), it it has a k-variable term satisfying these equations. It has been conjectured that a finite idempotent algebra A has finite relational width if and only if V(A) (the variety generated by A) has meet semi-distributive congruence lattices. The concept of “finite relational width” arises in the theory of complexity of algorithms, in the algebraic study of constraint-satisfaction problems. Actually, there are several different definitions of this concept and it is not known if they are equivalent. One version of the concept and the conjecture mentioned above are due to B. Larose and L. Zadori [10]. The important family of varieties with meet semi-distributive congruence lattices has various known characterizations. There is a characterization by a certain Maltsev condition; also, it is known that a locally finite variety has this property iff it omits congruence covers of types 1 and 2 (defined in the tame congruence theory of D. Hobby, R. McKenzie [6]). E. Kiss showed that a finite idempotent algebra of relational width k must have an m-WNU term operation for every m ≥ k. E. Kiss and M. Valeriote then observed that a finite algebra with a k-WNU term operation, k > 1, must omit congruence covers of type 1. These observations led M. Valeriote to make two conjectures: any locally finite variety omits congruence covers of type 1 iff it satisfies WNU(k) for some k > 1; any locally finite variety has meet semi-distributive congruence lattices if and only if for some k, it satisfies WNU(m) for all m ≥ k. In this paper, we prove both of these conjectures of M. Valeriote. The family of locally finite varieties omitting type 1 is the largest family of locally finite varieties defined by a nontrivial idempotent Maltsev condition. For this

Semisimple radical classes of involution algebras

J. Wichman has asked about semisimple radical classes of involution algebras. In the present paper we describe the semisimple radical classes of involution algebras over a field K* with involution *. If K* is infinite, then there are only trivial semisimple radical classes. If K* is finite then these classes are subdirect closures of strongly hereditary finite sets of finite idempotent algebras. In proving this result we determine the structure of certain simple involution algebras. We prove that the variety of symmetric involution algebras over Z(2) does not have attainable identities, answering a problem posed by Gardner [2]. Most of the results are valid also for involution rings (over the integers).

On the composition of idempotent functions

W. Sierpifiski has shown [1] that given any set A and any function f:An~A, f can be obtained by an appropriate composition of binary functions. In this paper we consider the corresponding problem for idempotent functions; f.'A"~A is idempotent iff (a, a ..... a)=a for all aeA. The motivation for this investigation is a sequence of papers by G. Gratzer and J. Plonka [2]-[5] characterizing the number p, of n-ary polynomials in an algebra. Satisfactory answers are known for all but idempotent algebras (an algebra is idempotent if each of its operations is idempotent). For a finite idempotent algebra it is easy to calculate the maximum asymptotic rate of growth of thepn'S and it is clear this rate is achieved when all idempotent functions are polynomials. Can this rate be achieved by an idempotent algebra with only a finite number of operations? Obviously the answer is yes if every idempotent function can be obtained by an appropriate composition of, say, binary idempotent functions. Let IAI be the cardinality of A. We shall now prove the following theorem: If IAI>2 then every idempotent function can be obtained by composition of binary idempotent functions; if IAI =2 then every idempotent function can be obtained by composition of ternary idempotent functions but not by binary idempotent functions. LEMMA 1: If IAI ~>No then for all n>~O every n-ary idempotent function can be obtained by composition of ternary idempotent functions.