Free l-groups

Abstract

From well-known results in universal algebra each variety or quasivariety K of algebraic systems has a free algebraic system F with the set X = {x _i |i ∈ I } of generators such that if φ̂ is a map of X into an algebraic system G ∈ K, then there is a homomorphism φ from F into G extending φ̂ (see the book of Burris and Sankappanavar [1], Theorem 11.4). The set X is called in this case the set of free generators. Therefore, each l-variety 𝓅 has a free l-group in 𝓅. If 𝓅 = ℒ, then a free l-group in ℒ is a called free l-group. The cardinality of the set of free generators of a free l-group in l-variety is called a rank of a free l-group. It is clear that a free l-group is a free l-group over trivially ordered free group (see Section 1 of Chapter 7). Let F _l (n) be a free l-group with free generators x ₁, …, x _n and Φ = {w _i (X) = w _i (x ₁, …, x _n )|i ∈ I} be a set of elements of F _l (n), i.e., the set of words of the signature l (l-words) in the variables x ₁, …, x _n. Let G be an l-group. Then the convex l-subgroup H generated by all substitutions of w _i (x ₁, …, x _n) in G is called a verbal l-subgroup of G. A convex l-subgroup H of G is called a fully invariant l-subgroup of G if Hφ ⊆ H for each l-homomorphism φ of G into G.