On steps of solubility of lattices and degrees of idempotency of prevarieties of lattices

V. B. Lender

Sbornik. Mathematics, Tome 24 (1974) no. 3, pp. 435-449

Résumé

Let $\mathfrak R$ be a sub-prevariety of a fixed prevariety (residually closed class) $\mathfrak U$. The smallest ordinal number $\gamma$ such that an algebra $A\in \mathfrak U$ is $\gamma$-step $\mathfrak R$-soluble is called the step of $\mathfrak R$-solubility of $A$. The smallest ordinal number $\eta\ne1$ such that there exists an $\eta$-step $\mathfrak R$-soluble algebra $A\in\mathfrak U$ is called the degree of idempotency of $\mathfrak R$ relative to $\mathfrak U$. In the paper $\mathfrak U$ is taken to be the class of all lattices, and all ordinal numbers that can be degrees of idempotency of prevarieties of lattices are found. Further, a description is given, depending on the degree of idempotency of a prevariety $\mathfrak R$, of the ordinal numbers that can be steps of $\mathfrak R$-solubility of suitable lattices. Bibliography: 11 titles.