关于几类2阶ai-半环生成的簇的研究

借助半环的格林关系研究了由所有2阶ai-半环生成的半环簇S2的一些子簇．其次,定义了与S2中半环的元素相关的同余关系,并揭示了同余关系之间的联系．     

关于一类半环上的格林关系的若干研究

研究了加法半群是带,乘法半群是完全正则半群的半环上的格林关系,给出了˙L∧+D(+L,+R)是同余关系的充分必要条件,证明了由这些同余关系所决定的半环类都是半环簇,并给出了这些半环簇的Mal′cev积分解.     

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．     

一类广义正则半环上可除半环同余的等价刻画

研究了一类广义正则半环的理想,利用这类理想,得到了这类半环上同余的几种刻画.     

一类循环系的平坦性

设Ｓ是幺半群,ｘ_１,ｘ₂,…∈Ｓ且满足ｘ_i+１ｘ_i＝ｘ_i,ｉ＝１,２……ｙ是S中的任意元素,记H={(yx_i,x_i)|i=1,2,…}.设ρ(H)是S的由Ｈ生成的最小右同余,本文证明了S／ρ（Ｈ）是平坦右Ｓ－系．     

关于半环上格林关系的开同余

首先给出了由半环的乘法半群上的格林关系所确定的半环开同余的性质和刻画.其次,由开同余出发,得到了六个不同的半环类,并证明了这六个半环类均是半环簇.最后,对半环簇的子簇格上的开算子进行了探讨,得到了一些有趣的结果.     

关于拟正则半环上可除半环同余的注记

研究了拟正则半环上的同余,给出了拟正则半环上一类理想的性质,并利用这类理想给出了拟正则半环上可除半环同余的几种等价形式.     

半环上同余的刻画

引入半环的完全素理想的概念.利用这一概念,研究了半环的同余,分别给出了半环的子直积分解的若干结果和半环同余的一种刻画.     

某些完全正则半环的刻画

研究了完全正则半环的特征.利用半群的方法,得到了当分配半环的乘法幂等元集分别是左零带、矩形带以及正规带时,该类半环成为完全正则半环的等价刻画,推广并改进了相关文献的主要结果.     

半环的范德瓦尔登问题

首先从半群理论角度解释了整环的分式域过程.其次,给出了构造给定半环的格罗滕迪克环的方法,进一步证明了任意加法可消半环可嵌入其格罗滕迪克环.最后,证明了半环上的同余与其格罗滕迪克环理想之间可以建立一一对应关系.     

The variety generated by equivalence algebras

Every equivalence relation can be made into a groupoid with the same underlying set if we define the multiplication as follows: xy = x if x,y are related; otherwise, xy = y. The groupoids, obtained in this way, are called equivalence algebras. We find a finite base for the equations of equivalence algebras. The base consists of equations in four variables, and we prove that there is no base consisting of equations in three variables only. We also prove that all subdirectly irreducibles in the variety generated by equivalence algebras are embeddable into the three-element equivalence algebra, corresponding to the equivalence with two blocks on three elements. Received September 21, 1998; accepted in final form May 11, 1999.     

The variety of commutative additively and multiplicatively idempotent semirings

The variety \({\mathcal Z}\) of commutative additively and multiplicatively idempotent semirings is studied. We prove that \({\mathcal Z}\) is generated by a single subdirectly irreducible three-element semiring and it has a canonical form for its terms. Hence, \({\mathcal Z}\) is locally finite despite the fact that it is residually large. The word problem in \({\mathcal Z}\) is solvable.     

The variety generated by planar modular lattices

We investigate the variety generated by the class of planar modular lattices. The main result is a structure theorem describing the subdirectly irreducible members of this variety.     

On a variety of commutative multiplicatively idempotent semirings

We prove that the variety \({{\mathscr {V}}}\) of commutative multiplicatively idempotent semirings satisfying \(x+y+xyz\approx x+y\) is generated by a single three-element semiring. Moreover, we describe a normal form system for terms in \({{\mathscr {V}}}\) and we show that the word problem in \({{\mathscr {V}}}\) is solvable. Although \({{\mathscr {V}}}\) is locally finite, it is residually big.     

Torsion and divisibility in finitely generated commutative semirings

It is conjectured that (additive) divisibility is equivalent to (additive) idempotency in a finitely generated commutative semiring S. In this paper we extend this conjecture to weaker forms of these properties—torsion and almost-divisibility (an element \(a\in S\) is called almost-divisible in S if there is \(b\in \mathbb {N}\cdot a\) such that b is divisible in S by infinitely many primes). We show that a one-generated semiring is almost-divisible if and only if it is torsion. In the case of a free commutative semiring F(X) we characterize those elements \(f\in F(X)\) such that for every epimorphism \(\pi \) of F(X) torsion and almost-divisibility of \(\pi (f)\) are equivalent in \(\pi (F(X))\).     

某些半环上Green关系的刻划

假设S是乘法半群为完全正则半群的半环.给出了S上的Green关系H,L和D是S上的半环同余的等价刻划,并利用幂等元的方法证明了在一定条件下D是S上的同余当且仅当L,R是S上的同余.     

On the variety generated by Murskii's algebra

It is shown that the variety generated by Murskii's algebra contains uncountably many subvarieties.Presented by K. A. Baker.