On Congruences on Ultraproducts of Algebraic Structures
Attila Nagy
TL;DR
This paper addresses how congruences behave under ultraproducts and ultrapowers of similar algebraic structures. It establishes an embedding theorem showing that the ultraproducts of component congruences embed into the congruence lattice of the ultraproduct, and an isomorphism theorem linking quotients by component congruences to ultraproducts of quotients. It specializes to ultrapowers, providing an explicit expression for restricted base-algebra congruences as joins of meets over partitions of the index set. Collectively, these results clarify the lattice-theoretic behavior of congruences in ultrafilter constructions and supply concrete tools for computing congruences in ultraproducts.
Abstract
Let $I$ be a non-empty set and $\mathcal{D}$ an ultrafilter over $I$. For similar algebraic structures $B_i$, $i\in I$ let $Π(B_i|i\in I)$ and $Π_{\mathcal{D}}(B_i|i\in I)$ denote the direct product and the ultraproduct of $B_i$, respectively. Let $\mathcal{D}^*$ denote the ultraproduct congruence on $Π(B_i|i\in I)$. Let the $\wedge$-semilattice of all congruences on an algebraic structure $B$ denoted by ${\bf Con}(B)$. In this paper we show that, for any similar algebraic structures $A_i$, $i\in I$, there is an embedding $Φ$ of $Π_{\mathcal{D}}({\bf Con}(A_i)|i\in I)$ into ${\bf Con}(Π_{\mathcal{D}}(A_i|i\in I)$. We also show that, for every $σ\in Π({\bf Con}(A_i)|i\in I)$, the factor algebra $Π_{\mathcal{D}}(A_i|i\in I)/Φ(σ/\mathcal{D}^*)$ is isomorphic to $Π_{\mathcal{D}}(A_i/σ(i)|i\in I)$. Moreover, if $A$ is an algebraic structure, $σ(i)\in {\bf Con}(A)$, $i\in I$ and $\mathcal{D}=\{ K_j| j\in J\}$ then the restriction of $Φ(σ/\mathcal{D}^*)$ to $A$ equals $\vee _{j\in J}(\wedge _{k\in K_j}σ(k))$.