Generating solutions of a linear equation and structure of elements of the Zelisko group II

Generating solutions of a linear equation and structure of elements of the Zelisko group

Solutions of a linear equation b=ax in a homomorphic image of a commutative Bezout domain of stable range 1.5 is developed. It is proved that the set of solutions of a solvable linear equation contains at least one solution that divides the rest, which is called a generating solution. Generating solutions are pairwise associates. Using this result, the structure of elements of the Zelisko group is investigated.

A Generalized Next-Closure Algorithm -- Enumerating Semilattice Elements from a Generating Set

We present a generalization of the well known Next-Closure algorithm working on semilattices. We prove the correctness of the algorithm and apply it on the computation of the intents of a formal context.

分块格四元素生成集中的原子

Atoms in four-element generating sets of partition lattices

本文证明 Part(n) 中高度一、二的元素及原子均可纳入四元素生成集，并结合构造得到可数无限分块格的四生成性与 Part(n) 的大量生成集。

Generating Endomorphism Rings of Infinite Direct Sums and Products of Modules

Let R be a ring, M a left R-module, I an infinite set, N either the direct sum or product of |I| copies of M, and E the endomorphism ring of N as a left R-module. In this note it is shown that E is not the union of a chain of |I| or fewer proper subrings, and also that given a generating set U for E as a ring, there exists a positive integer n such that every element of E is represented by a ring word of length at most n in elements of U.

Four-element generating sets of partition lattices and their direct products

Let $n>3$ be a natural number. By a 1975 result of H. Strietz, the lattice Part$(n)$ of all partitions of an $n$-element set has a four-element generating set. In 1983, L. Zádori gave a new proof of this fact with a particularly elegant construction. Based on his construction from 1983, the present paper gives a lower bound on the number $ν(n)$ of four-element generating sets of Part$(n)$. We also present a computer assisted statistical approach to $ν(n)$ for small values of $n$. In his 1983 paper, L. Zádori also proved that for $n\geq 7$, the lattice Part$(n)$ has a four element generating se…

用少量元素生成布尔格与会话密钥交换

Generating Boolean lattices by few elements and exchanging session keys

本文证明了布尔格的最小生成集规模由 Sp(k) 刻画，并据此基于等价格协议提出一种会话密钥交换的密码方案，同时证明若窃听者能在多项式时间内攻破简化协议则 P=NP，并附带得到有限布尔格中一类方程组可解性为 NP 完全的结果。