On Congruences on Ultraproducts of Algebraic Structures

TL;DR

This paper investigates the structure of congruences on ultraproducts of algebraic structures, establishing embeddings and isomorphisms that relate congruences on ultraproducts to those on component structures.

Contribution

It introduces an embedding of ultraproducts of congruence lattices into the congruence lattice of the ultraproduct, and characterizes the resulting quotient structures.

Findings

Embedding of ultraproduct congruences into the larger congruence lattice.

Isomorphism between certain quotient ultraproducts and ultraproducts of quotient structures.

Explicit description of the restriction of the embedding to algebraic structures.

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 $\Pi (B_i|i\in I)$ and $\Pi _{\mathcal{D}}(B_i|i\in I)$ denote the direct product and the ultraproduct of $B_i$, respectively. Let $\mathcal{D}^*$ denote the ultraproduct congruence on $\Pi (B_i|i\in I)$. 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 $\Phi$ of $\Pi _{\mathcal{D}}({\bf Con}(A_i)|i\in I)$ into ${\bf Con}(\Pi _{\mathcal{D}}(A_i|i\in I)$. We also show that, for every $\sigma \in \Pi ({\bf Con}(A_i)|i\in I)$, the factor algebra $\Pi _{\mathcal{D}}(A_i|i\in I)/\Phi (\sigma /\mathcal{D}^*)$ is isomorphic to $\Pi _{\mathcal{D}}(A_i/\sigma (i)|i\in I)$. Moreover, if $A$ is an algebraic structure, $\sigma(i)\in {\bf Con}(A)$, $i\in I$ and $\mathcal{D}=\{ K_j| j\in J\}$ then the restriction of $\Phi (\sigma /\mathcal{D}^*)$ to $A$ equals $\vee _{j\in J}(\wedge _{k\in K_j}\sigma (k))$.