Existentially closed structures and some embedding theorems

Abstract

Using the notion of existentially closed structures, we obtain embedding theorems for groups and Lie algebras. We also prove the existence of some groups and Lie algebras with prescribed properties.

Existentially closed Leibniz algebras and an embedding theorem

“…In this section we provide an embedding theorem. To this end, we use the following lemma and theorem proved by Shahryari in [11]. In fact, both lemma and theorem can be considered for an arbitrary non-associative algebra.…”

“…Lemma 1. [11]. Let V be an inductive class of algebras over a field K. Suppose V is closed under subalgebra and L ∈ V. Then there exists an algebra H ∈ V containing L such that its dimension is at most max{ℵ 0 , dim L, |K|}.…”

“…Having considered the properties of closure of algebraic systems in both existentially and algebraically senses, we can claim that they are equivalent concepts for groups and Lie algebras. We recall that an algebraic system A is existentially closed, if every consistent finite set of existential sentences with parameters from A, is satisfiable in A. Shahryari in [11] used the concept of existentially closed groups and Lie algebras to prove some embedding theorems. For instance, Shahryari showed that any Lie algebra L can be embedded in a simple Lie algebra in such a way for any non-zero elements a and b, there is x such that [x, a] = b.…”

On Strongly Algebraically Closed Lattices

“…In a preprint, M. Shahryari investigated some applications of these concepts to special classes of groups (see [5]). In this paper, we focus on distributive lattices.…”

“…The organization of the paper is as follows: in Section 1, we show that if a class X of lattices is inductive and closed under elementary sublattices, then every element of X has an extension which is existentially closed in X. In fact, this result is not new and at least a version of it for classes of groups is presented in [5]. However, in our version, the assumption of being closed under elementary substructures is applied instead of the stronger hypothesis of being closed under substructures.…”