On Congruences on Ultraproducts of Algebraic Structures

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 $\vee _{j\in J}(\wedge _{k\in K_j}\sigma (k))$.