On right congruences of semigroups having no proper essential right congruences

A right congruence ?? in a semigroup S is essential if for any right congruence ?? we have ??????=?? (the identity relation) implies ??=??. Clearly, the universal relation, ??, is an essential right congruence. We say ?? is proper if ??????. In this paper we get a necessary and sufficient condition for a semigroup with an identity element?1 and having no proper essential right congruences to have a distributive lattice of right congruences.     

Finitely nonstationary nondeterministic automata with random input

Properties of generalized finitely nonstationary nondeterministic automata with an additional random input over a Boolean lattice are considered which are related to the definition of the class of languages represented by such automaton models. New notions of an elementary nondeterministic automatic structure with a random input, of a generalized finitely nonstationary nondeterministic automaton with a random input, of the generalized mapping induced by such an automaton, and of a generalized language represented by such an automaton are introduced. A number of statements substantiating synthesis for any given generalized finitely nonstationary nondeterministic automaton with a random input of an abstract probabilistic finite automaton equivalent to the given one relative to the represented generalized language probabilistic language of the stationary abstract probabilistic finite automaton. The number of states of the synthesized probabilistic automaton is estimated and a synthesis algorithm is developed in detail and illustrated by an example.     

The direct product of right congruences

In a number of earlier papers the study of the structure of semigroups has been approached by means of right congruences. Such an approach seems appropriate since a right congruence is one of the possible analogs of both the right ideal of a ring and the subgroup in a group. Each of these substructures plays a strong role in the study of the structure of their respective systems. In both the ring and the group the internal direct product is naturally, and effectively, defined. However, what such an internal direct product should be for two right congruences of a semigroup is not so clear. In this paper we will offer a possible definition and consider some of the consequences of it. We will also extend some of these results to automata.     

On polynomial congruences

The work of the first author was supported by the Russian Foundation for Fundamental Research, Grant 93-011-240.     

Structure of digraphs associated with quadratic congruences with composite moduli

We assign to each positive integer n a digraph G(n) whose set of vertices is H={0,1,…,n-1} and for which there exists a directed edge from a∈H to b∈H if . Associated with G(n) are two disjoint subdigraphs: G₁(n) and G₂(n) whose union is G(n). The vertices of G₁(n) correspond to those residues which are relatively prime to n. The structure of G₁(n) is well understood. In this paper, we investigate in detail the structure of G₂(n).     

On multiplicative congruences

Let ${\varepsilon}$ be a fixed positive quantity, m be a large integer, x _j denote integer variables. We prove that for any positive integers N ₁, N ₂, N ₃ with ${N_1N_2N_3 > m^{1+\varepsilon}, }$ the set $$\{x_1x_2x_3 \quad ({\rm mod}\,m): \quad x_j\in [1,N_j]\}$$ contains almost all the residue classes modulo m (i.e., its cardinality is equal to m + o(m)). We further show that if m is cubefree, then for any positive integers N ₁, N ₂, N ₃, N ₄ with ${ N_1N_2N_3N_4 > m^{1+\varepsilon}, }$ the set $$\{x_1x_2x_3x_4 \quad ({\rm mod}\,m): \quad x_j\in [1,N_j]\}$$ also contains almost all the residue classes modulo m. Let p be a large prime parameter and let ${p > N > p^{63/76+\varepsilon}.}$ We prove that for any nonzero integer constant k and any integer ${\lambda\not\equiv 0 \,\, ({\rm mod}\,p)}$ the congruence $$p_1p_2(p_3+k)\equiv \lambda \quad ({\rm mod}\, p) $$ admits (1 + o(1))π(N)³/p solutions in prime numbers p ₁, p ₂, p ₃ ≤ N.     

Semigroups whose lattices of right congruences are semiatomic

The main theorems presented here are characterizations of a semigroup with a left identity whose lattice of right congruences is semiatomic. These theorems are preceded by a number of results on minimal right congruences.     

On the congruences of Jacobi forms

In this paper, we study congruence properties of coefficient of Jacobi forms. The result for elliptic modular form case was studied by Sturm (Lecture Notes in Mathematics, Springer, Berlin Heidelberg New York 1987).     

On the sectional irregularity of congruences

Partially supported by CICYT PB90-0637     

On congruences of m-groups

We indicate a way for constructing m-congruences of an arbitrary m-transitive representation, introduce the notions of m-2-transitive and m-primitive representations, and describe the m-transitive primitive representations in terms of stabilizers. Also we give necessary and sufficient conditions for m-2-transitivity and study some properties of these representations.     

On congruences of groupoids closely connected with quasigroups

Conditions when a congruence of a left (right) division groupoid and a left (right) cancellation groupoid is closed (“normal”) are given. Conditions for the simplicity of the above-mentioned groupoids are obtained.     

Partition identities and congruences associated with the Fourier coefficients of the Euler products

In this article, we discuss two applications of the operator U(m) (see (1.1)) defined on the product of two power series.     

On congruences of Jacobi forms

We consider congruences and filtrations of Jacobi forms. More specifically, we extend Tate's theory of theta cycles to Jacobi forms, which allows us to prove a criterion for an analog of Atkin's -operator applied to a Jacobi form to be nonzero modulo a prime.

     

On the structure of varieties with equationally definable principal congruences III

p(x, y, z) is aternary deduction (TD) term function on an algebra A if, for alla, b A, p(a, b,z) z (mod (a, b)), and {p(a, b, z): z A} is a transversal of the set of equivalence classes of the principal congruence (a, b). p iscommutative ifp(a, b, z) and p(a', b', z) define the same transversal whenever0(a, b)=0(a', b'). p isregular if(p(x, y, 1), 1)=0(x, y) for some constant term 1. The TD term generalizes the (affine) ternary discriminator and is used to investigate the logical structure of nonsemisimple varieties with equationally definable principal congruences (EDPC). Some of the results obtained: The following are equivalent for any variety: (1)V has a TD term; (2)V has EDPC and a certain strong form of the congruence-extension property. IfV is semisimple and congruence-permutable, (1) and (2) are equivalent to (3)V is an affine discriminator variety. Afixedpoint ternary discriminator on a set is defined by the conditions:p(x, x,z)=z and, ifx y, p(x, y,z)=d whered is some fixed element; afixedpoint discriminator variety is defined in analogy to affine discriminator variety. The commutative TD term generalizes the fixedpoint ternary discriminator. The following are equivalent for any semisimple variety: (4)V has a commutative TD term; (5)V is a fixedpoint discriminator variety. IfV is semisimple, congruence-permutable, and has a constant term, (4) and (5) are equivalent to (3); ifV has a second constant term distinct from the first in all nontrivial members ofV then all five conditions are equivalent to (6)V has a commutative, regular TD term. Ahoop is a commutative residuated monoid.Hoops with dual normal operators are defined in analogy with normal Boolean algebras with operators. The main result: A variety of hoops with dual normal operators has a commutative, regular TD term iff it has EDPC iff it has first-order definable principal congruences.Dedicated to Bjarni Jónsson on his 70th birthdayPresented by R. W. Quackenbush.The authors gratefully acknowledge the support of National Science Foundation Grants DMS-8703743 and DMS-8805870.     

On the structure of varieties with equationally definable principal congruences IV

The notion of apseudo-interior algebra is introduced; it is a hybrid of a (topological) interior algebra and a residuated partially ordered monoid. The elementary arithmetic of pseudo-interior algebras is developed leading to a simple equational axiomatization. A notion ofopen filter analogous to the open filters of interior algebras is investigated. Pseudo-interior algebras represent, in algebraic form, the logic inherent in varieties with acommutative, regular ternary deductive (TD) term p(x, y, z), which is defined by the conditions: (1)p(x,y,z) z (mod(x, y)); (2) for fixed elementsa, b of an algebra A, {p(a, b, z):z A} is a transversal of the set of equivalence classes of (a, b); (3)p(a, b, z) andp(a,b,z) define the same transversal whenever(a,b)=(a,b); (4)(p(x, y, 1), 1)= (x, y) for some constant term 1. The TD term generalizes the (affine) ternary discriminator. Varieties with a commutative, regular TD term include most of the varieties of traditional algebraic logic as well as all double-pointed affine discriminator varieties andn-potent hoops (residuated commutative po-monoids in which the partial ordering is inverse divisibility). The main theorem:A variety has a commutative, regular TD term iff it is termwise definitionally equivalent to a pseudo-interior algebra with additional operations that are compatible with the open filters in a natural way.Presented by R. W. Quackenbush.The authors gratefully acknowledge the support of National Science Foundation Grants DMS-8703743 and DMS-8805870.