Σ-结构的Σ-自由积

FREE Σ-PRODUCT OF Σ-STRUCTURES

本文的目的是把代数的自由积的概念推广到结构的Σ-自由积,部分的回答了[1]中提出的第94,95,97三个问题。定义1.设K是Σ-结构的一个类,u是一个部分结构。那么称F_k(u)是由部分结构u在K内Σ-自由生成的一个Σ-结构,如果下面的条件成立:

This paper partly solved problems 94, 95, 97 in Universal Algebra of G. Gratzer. If φ is a mapping of AintoB, then φ is called a homomorphism of Uinto Bif it is homomorphism of <A;F> into <B;F> and r_γ (a_0,…,a_m_γ^(-1)), a_0,…, a_m_γ^(-1)∈A, imply that r_γ(a_0φ,…,a_m_γ^(-1)φ) and if r_γ(b_0,…,b_m_γ^(-1)), b_0, …, b_m_γ^(-1)∈B, imply the existence of a_0, …, a_m_γ^(-1)∈A such that r_γ(a_0, …, a_m_γ^(-1)) and a_0φ=b_0, …,a_m_γ^(-1) φ=b_m_y^(-1). Theorem 1. LetKbe a class of Σ-structures and letube a Σ-structure. Assume that the following conditions hold: (ⅰ) F_∑(m) exists for some m≧|A|and F_∑(m)∈K; (ⅱ) H(K)K; (ⅲ) for any B∈K, ifφis a homomorphism of U intoB, then φis a∑-homomorphism of UintoB. then F_∑(U) exists and F_k(U)≌(U). Theorem 3. The free ∑-product of U_i, i∈I, in the class K of pratial structures exists if and only if(*)and the following conditions are satisfied: (ⅰ) L has a maximal homomorphism image L in K; let x denote homomorphism of L onto; (ⅱ) if a, b∈A, a≠b, then ax_ix≠bx_ix; (ⅲ) F_k exists.