Congruence modularity implies cyclic terms for finite algebras

An n-ary operation f : A ⁿ → A is called cyclic if it is idempotent and ${f(a_1, a_2, a_3, \ldots , a_n) = f(a_2, a_3, \ldots , a_n, a_1)}$ for every ${a_1, \ldots, a_n \in A}$ . We prove that every finite algebra A in a congruence modular variety has a p-ary cyclic term operation for any prime p greater than |A|.     

Pappian implies Arguesian for modular lattices

We prove the claim in the title, using Hessemberg's analog theorem for projective planes, and the characterization of non-Arguesian modular lattices provided by Day and Jónsson [5,6]. Received January 22, 1999; accepted in final form May 6, 1999.     

Two notes on the Arguesian identity

We find an explicitly self-dual lattice identity equivalent to the Arguesian law. We also show that any lattice identity equivalent to the Arguesian law must necessarily involve at least six variables.Presented by Alan Day.     

Congruence modularity at 0

Given a variety ${\mathcal{V}}$ with a constant 0 in its type and a lattice identity p ?? q, we say that p ?? q holds for congruences in ${\mathcal{V}}$ at 0 if the p-block of 0 is included in the q-block of 0 for all substitutions of congruences of ${\mathcal{V}}$ -algebras for the variables of p and q. Varieties that are congruence modular at 0 are characterized by a Mal??tsev condition. This result generalizes the classical characterization of congruence modularity by Day terms.     

Semiregularity of congruences implies congruence modularity at 0

We introduce a weakened form of regularity, the so called semiregularity, and we show that if every diagonal subalgebra of is semiregular then is congruence modular at 0.     

Congruence lower semimodularity and 2-finiteness imply congruence modularity

We show that any congruence lower semimodular variety whose 2-generated free algebra is finite must be congruence modular.Presented by Walter Taylor.     

Congruence join semidistributivity is equivalent to a congruence identity

We show that a locally finite variety is congruence join semidistributive if and only if it satisfies a congruence identity that is strong enough to force join semidistributivity in any lattice. Received February 9, 2000; accepted in final form November 23, 2000.     

On Automorphism Groups of Arguesian Lattices

Every (finite) group is isomorphic to the automorphism group of some (finite) subdirectly irreducible Arguesian lattice.     

On the congruence modularity conjecture

A new condition of compatibility with projections, applicable to some Maltsev filters, is defined and shown to hold, among others, for the filter of congruence-modular varieties. As a consequence, it is shown that there exist no simple counterexamples (in a specified sense) to the modularity conjecture. This paper is dedicated to Walter Taylor. Received November 5, 2005; accepted in final form April 3, 2006.     

Randomness implies order

Let τ: [0, 1] → [0, 1] possess a unique invariant density $\text{f}{}^1$. Then given any ? > 0, we can find a density function p such that $\text{∥ p ? f}{}^1\text{∥ < ?,}\text{and}\text{p}$ is the invariant density of the stochastic difference equation x_{n + 1} = τ(x_n) + W, where W is a random variable. It follows that for all starting points $\text{x}{}_0\text{? [0, 1],}\text{lim}{}_{\mathrm{n\to \infty }}\text{(1}\text{n)}\text{∑}{}_{\mathrm{i\; =\; 0}}{}^{\mathrm{n\; ?\; 1}}\text{χ}{}_{\mathrm{B}}\text{(x}{}_{\mathrm{i}}\text{) = ∝}{}_{\mathrm{B}}\text{p(ξ) dξ}$.     

有关Lucas与Babbage同余式的推广

利用一个素数模意义上的整数分拆的结果,证明了高斯二项式系数上的Lucas同余式,并且分别研究了Lucas同余式和Babbage同余式在广义二项式系数上的情况,得到了相应的形式简洁的同余式.     

Narrowness implies uniformity

This paper is about varietiesV of universal algebras which satisfy the following numerical condition on the spectrum: there are only finitely many prime integersp such thatp is a divisor of the cardinality of some finite algebra inV. Such varieties are callednarrow. The variety (or equational class) generated by a classK of similar algebras is denoted by V(K)=HSPK. We define Pr (K) as the set of prime integers which divide the cardinality of a (some) finite member ofK. We callK narrow if Pr (K) is finite. The key result proved here states that for any finite setK of finite algebras of the same type, the following are equivalent: (1) SPK is a narrow class. (2) V(K) has uniform congruence relations. (3) SK has uniform congruences and (3) SK has permuting congruences. (4) Pr (V(K))= Pr(SK). A varietyV is calleddirectly representable if there is a finite setK of finite algebras such thatV= V(K) and such that all finite algebras inV belong to PK. An equivalent definition states thatV is finitely generated and, up to isomorphism,V has only finitely many finite directly indecomposable algebras. Directly representable varieties are narrow and hence congruence modular. The machinery of modular commutators is applied in this paper to derive the following results for any directly representable varietyV. Each finite, directly indecomposable algebra inV is either simple or abelian.V satisfies the commutator identity [x,y]=x·y·[1,1] holding for congruencesx andy over any member ofV. The problem of characterizing finite algebras which generate directly representable varieties is reduced to a problem of ring theory on which there exists an extensive literature: to characterize those finite ringsR with identity element for which the variety of all unitary leftR-modules is directly representable. (In the terminology of [7], the condition is thatR has finite representation type.) We show that the directly representable varieties of groups are precisely the finitely generated abelian varieties, and that a finite, subdirectly irreducible, ring generates a directly representable variety iff the ring is a field or a zero ring.