Finite Universal Algebra Research Articles

О синтаксическом определении класса языков, распознаваемых недетерминированными машинами Тьюринга на логарифмической памяти

NL is defined as the class of languages recognizable by logspace nondeterministic Turing machines. One of the main unsolved problems in complexity theory is that of relation between classes P and NL. It is known that NL is contained in P, since there is a polynomial-time algorithm for 2-satisfiability, but it is not known whether NL = P or whether NL = L. A possible approach to these problems can be based on searching for an alternative suitable definition of the class NL. Taitslin et al. propose such a definition in terms of nondeterministic programs. The syntax of such programs is similar to that of usual computer programs. Each nondeterministic program takes a finite universal algebra as input. Taitslin et al. defined a class of languages recognizable by such programs and proved that this class is a subclass of NL. In the present paper, we slightly modify their syntactical definition. Namely, we modify a definition of nondeterministic program input and give a new definition of a language recognized by a given program. We prove that the class of languages recognizable by nondeterministic programs according to our definition is just NL.

On the role of logical connectives for primality and functional completeness of algebras of logics

We consider certain finite universal algebras arising from algebraic semantics in implicational logics. They contain a binary operation -> and two constants 0 and 1 satisfying the axioms 0->x=x->1=x->x=1 and 1->x=x valid in most implicational logics. We characterize the completeness (also called primality) of such algebras, i.e. the property that every finitary operation on their universe is a term operation of the algebra (in other words, it is a composition of the basic operations of the algebra). Using clone theory and the knowledge of maximal clones we describe completeness (functional completeness) in terms of nonpreservation of three types of specific relations. If -> has a simple property and the algebra contains a binary operation @? with a neutral element 1 and a unary operation @? satisfying @?(1)=0 and @?x=1 otherwise, the algebra is functionally complete.

Classifying the Complexity of Constraints Using Finite Algebras

Many natural combinatorial problems can be expressed as constraint satisfaction problems. This class of problems is known to be NP-complete in general, but certain restrictions on the form of the constraints can ensure tractability. Here we show that any set of relations used to specify the allowed forms of constraints can be associated with a finite universal algebra and we explore how the computational complexity of the corresponding constraint satisfaction problem is connected to the properties of this algebra. Hence, we completely translate the problem of classifying the complexity of restricted constraint satisfaction problems into the language of universal algebra. We introduce a notion of "tractable algebra," and investigate how the tractability of an algebra relates to the tractability of the smaller algebras which may be derived from it, including its subalgebras and homomorphic images. This allows us to reduce significantly the types of algebras which need to be classified. Using our results we also show that if the decision problem associated with a given collection of constraint types can be solved efficiently, then so can the corresponding search problem. We then classify all finite strictly simple surjective algebras with respect to tractability, obtaining a dichotomy theorem which generalizes Schaefer's dichotomy for the generalized satisfiability problem. Finally, we suggest a possible general algebraic criterion for distinguishing the tractable and intractable cases of the constraint satisfaction problem.

Frattini Theory for Classes of Finite Universal Algebras of Malcev Varieties

We extend the Frattini theory of formations and Schunck classes of finite groups to some Frattini theory of formations and Schunck classes of finite universal algebras of Malcev varieties. We prove that if F≠(1) is a nonempty formation (Schunck class) of algebras of a Malcev variety, then its Frattini subformation (Frattini Schunck subclass) Φ(F) consists of all nongenerators of F; moreover, if M is a formation (Schunck class) in F; then Φ(M)\( \subseteq\)Φ(F).

Conditional identity calculus and conditioned rational equivalence

We construct a conditional identity calculus (similar to the Birkhoff identity calculus), which complies with the concept of truth for a conditional identity on a universal algebra. The relationship is studied between the isomorphism of embedding categories of conditional varieties and the conditioned rational equivalence of these varieties. As applications, we describe invariants for the relations ‘is conditional rational equivalent’ and ‘is similar’ on finite universal algebras.

Finite simple abelian algebras are strictly simple

A finite universal algebra is called strictly simple if it is simple and has no nontrivial subalgebras. An algebra is said to be Abelian if for every term t(x,yp) and for all elements a, b, c, d, we have the following implication: t(a, c) = t(a, d) -t(b, c) = t(b, d) . It is shown that every finite simple Abelian universal algebra is strictly simple. This generalizes a well-known fact about Abelian groups and modules.

INHERENTLY NONFINITELY BASED FINITE SEMIGROUPS

A locally finite variety is called inherently nonfinitely based if it is not contained in any finitely based locally finite variety. A finite universal algebra is called inherently nonfinitely based if it generates an inherently nonfinitely based variety. In this paper a description of inherently nonfinitely based finite semigroups is given; it is proved that the set of such semigroups is recursive and that the property of a finite semigroup to be inherently nonfinitely based is mainly determined by the structure of its subgroups. It is also shown that there exists a unique minimal inherently nonfinitely based variety of semigroups consisting not only of groups. It is not known whether there exists an inherently nonfinitely based variety of groups.Bibliography: 18 titles.

Representations of finite universal algebras in finite semigroups

An answer is given to the question of what conditions must be imposed on a signature in order that every finite algebra with this signature can be represented in the Mal'tsev-Cohn sense in a finite semigroup.

Boolean extensions and normal subdirect powers of finite universal algebras

(i) ~B is subdirectly representable in {91o . . . . ,91,_2}c_K if and only if satisfies the identities of 91o x ... x 91n2. (ii) I f I f3 ] > 1 and ~ is subdirectly representable in K, then ~ has a unique set of factors in K. We shall generalize the concept of normal subdirect power in a way that will enable us to prove Theorems 1 * and 2* for arbi t rary (rather than binary or f-) algebras; Theorem 3 will be proved without recourse to the theory of f algebras.