Congruence Kernels in Weakly Regular Varieties

We generalize the concept of deductive system of Hilbert algebras for the general case of algebras in weakly regular variety and we show that congruence kernels of these algebras are just deductive systems. This concept enables us to describe congruence classes in algebras of weakly regular varieties.AMS Subject Classification (1991): 08A30 08B05     

Some Ideal Lattices in Partial Abelian Monoids and Effect Algebras

Congruences and ideals in partial Abelian monoids (PAM) are studied. It is shown that the so-called R ₁-ideals in cancellative PAMs (CPAM) form a complete Brouwerian sublattice of the lattice of all ideals, and they are standard elements of it. In a special class of CPAMs, effect algebras, properties of ideals and congruences are studied in relation to the generalized Sasaki projections and dimensional equivalence.     

On complete congruence lattices of join-infinite distributive lattices

In [5] G. Gr?tzer and E. T. Schmidt raised the problem of characterizing the complete congruence lattices of complete lattices satisfying the Join-Infinite Distributive Identity (JID) and the Meet-Infinite Distributive Identity (MID) and proved the theorem: Any complete lattice with more than two elements and with a meet-irreducible zero cannot be represented as the lattice of complete congruence relations of a complete lattice satisfying the (JID) and (MID). In this note we generalize this result by showing that the complete congruence lattice of every complete lattice satisfying (JID) and (MID) is a zero-dimensional complete lattice satisfying (JID). Some consequences are discussed. Received March 6, 2000; accepted in final form September 12, 2000.     

有限秩的可解群的剩余有限性质

本文讨论了有限秩的可解群的剩余有限性质,推广了Smelkin等人关于多重循环群的几个同类结果．     

Congruences of Clone Lattices, II

We continue the study of congruences of clone lattices _A , where A is finite, started in an earlier paper by the author and A. P. Semigrodskikh. We prove that each clone that either contains all unary operations or consists of essentially unary operations forms a one-element class of any non-trivial congruence of _A . As a consequence, we get that _A has the greatest non-trivial congruence provided the lattice is not simple, that _A is directly indecomposable, and that it has neither distributive nor dually distributive elements except for the trivial ones.For |A|>2, no example of a non-trivial congruence is known so far. We exhibit some reasons why such congruences are not easy to find.     

On a matrix representation of a free group

An analog of the well-known Sanov representation of a free non-Abelian group by matrices of size 3 is studied. Instead of transvections used in the Sanov representation, we use matrices with filled first (second, etc.) column, except for the intersection with the diagonal, and we have ones on the diagonal and zeros at the other places. The filled places are occupied by the same parameterk. It is proved that, for ¦k¦5, these matrices generate a free group. However, fork=2, this is not the case.Translated fromMatematicheskie Zametki, Vol. 64, No. 6, pp. 863–870, December, 1998.     

Congruences and ideals in ternary rings

A ternary ring is an algebraic structure R=(R,t0.1) of type (3, 0, 0) satisfying the identities t(0, x, y) = y = t(x, 0, y) and t(1, x, 0) = x = (x, l, 0) where, moreover, for any a, b, c R there exists a unique d R with t(a, b, d) = c. A congruence on R is called normal if R with t is a ternary ring again. We describe basic properties of the lattice of all normal congruences on R and establish connections between ideals (introduced earlier by the third author) and congruence kernels.     

关于分圆多项式系数的注记

自然数m称为HΓ数,如果分圆多项式Fm(x)的系数只能是0或±1.本文研究自然数m成为HΓ数的条件,证明:如果p是素数,那么1)m=15p(p>5)为HΓ数的充分必要条件是p≡±1(mod30);2)m=21p(p>7)为HΓ数的充分必要条件是p≡±1,±11,±19(mod42);3)m=33p(p>11)为HΓ数的充分必要条件是p≡±1,±7,±25(mod66).     

具有Clifford断面的正则纯正半群

给出了具有Clifford断面的右正规纯正半群的等价刻画,得到了具有Clifford断面的正则纯正半群的次直积分解,证明了具有Clifford断面的正则纯正半群一定是正则纯正群．     

一类广义正则半环上同余的特征

在半环中引入了一类理想的概念,讨论了这类理想的性质,并研究了一类广义正则半环上的同余,给出了这类半环上一种半环同余的特征．