On Congruences on Ultraproducts of Algebraic Structures
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arXiv
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| Formato: | Preprint |
| Publicado: |
2015
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| _version_ | 1866915523107749888 |
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| author | Nagy, Attila |
| author_facet | Nagy, Attila |
| contents | 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 $\vee _{j\in J}(\wedge _{k\in K_j}σ(k))$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1511_02467 |
| institution | arXiv |
| publishDate | 2015 |
| record_format | arxiv |
| spellingShingle | On Congruences on Ultraproducts of Algebraic Structures Nagy, Attila Category Theory Rings and Algebras 08A30, 03C20 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 $\vee _{j\in J}(\wedge _{k\in K_j}σ(k))$. |
| title | On Congruences on Ultraproducts of Algebraic Structures |
| topic | Category Theory Rings and Algebras 08A30, 03C20 |
| url | https://arxiv.org/abs/1511.02467 |