标题： 格的模和下模元素 显示英文标题

标题： Modular and lower-modular elements of lattices of semigroup varieties

摘要： 本文包含三个主要结果。 首先，我们证明，如果一个交换半群种类是所有交换半群种类格中的模元素，则它要么是所有交换半群的种类COM，要么是幂零种类，要么是幂零种类与半格种类的并。 其次，我们证明，如果一个交换幂零种类是Com中的模元素，则它只能通过0-简化的和可替换的恒等式在COM中给出。 第三，我们完全分类了Com中的所有下模元素。 作为推论，我们证明，当Com中的一个元素是下模元素时，它也是模元素。 所有这些结果都是先前由Jezek、McKenzie、Vernikov和作者针对所有半群种类格中的模元素和下模元素所得结果的精确类似结果。 作为本文中开发的一种技术的应用，我们提供了我们第一个和第三个结果的“原型”的新证明。 显示英文摘要

摘要： The paper contains three main results. First, we show that if a commutative semigroup variety is a modular element of the lattice Com of all commutative semigroup varieties then it is either the variety COM of all commutative semigroups or a nil-variety or the join of a nil-variety with the variety of semilattices. Second, we prove that if a commutative nil-variety is a modular element of Com then it may be given within COM by 0-reduced and substitutive identities only. Third, we completely classify all lower-modular elements of Com. As a corollary, we prove that an element of Com is modular whenever it is lower-modular. All these results are precise analogues of results concerning modular and lower-modular elements of the lattice of all semigroup varieties obtained earlier by Jezek, McKenzie, Vernikov, and the author. As an application of a technique developed in this paper, we provide new proofs of the `prototypes' of the first and the third our results.