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 Apr 4 comment Why don't we study algebraic objects with more than two operations? @ChrisEagle Why fields can not be treated as universal algebras? Apr 4 answered Why don't we study algebraic objects with more than two operations? Apr 4 accepted Are there rules in the useage of prepositions in Math? Mar 30 revised Are there rules in the useage of prepositions in Math? fix Mar 30 comment Are there rules in the useage of prepositions in Math? @TaraB I think that the 'form word' I expressed is 'syntactic expletive', but in precisely in this post they are indeed prepositions. Mar 30 revised Are there rules in the useage of prepositions in Math? fix Mar 30 comment Are there rules in the useage of prepositions in Math? Yes, you are right. Mar 30 asked Are there rules in the useage of prepositions in Math? Mar 27 comment What is the correspondence between the semantic compactness of logics and the compactness of Stone spaces? Okay, let $\mathbf 1=Cn(\emptyset)$ is well. Both $\mathfrak F(\mathcal L)$ and $\mathfrak F(\mathcal L)/ \Leftrightarrow$ can be complete lattice. Mar 27 revised What is the correspondence between the semantic compactness of logics and the compactness of Stone spaces? added 4 characters in body Mar 27 revised What is the correspondence between the semantic compactness of logics and the compactness of Stone spaces? rolled back to a previous revision Mar 27 comment What is the correspondence between the semantic compactness of logics and the compactness of Stone spaces? Unfortunately, it isn't, e.g. $\{p,q\} \Leftrightarrow \{p \land q\}$ and $\{q\} \Leftrightarrow \{q\}$, but $\{p,q\} \cap \{q\}=\{q\}$ whereas $\{p \land q\} \cap \{q\}=\emptyset$, they are not logical equivalent, thus $\Leftrightarrow$ is not a congruence again... So let me rollback the old definition, under that $\frak F(\mathcal L)/\Leftrightarrow$ can be sure a complete lattice. Mar 27 comment What is the correspondence between the semantic compactness of logics and the compactness of Stone spaces? But now is $\Leftrightarrow$ still a congruence? It's a question... Mar 27 comment What is the correspondence between the semantic compactness of logics and the compactness of Stone spaces? @MihaHabič Okay, thank you. I have rewritten (2') that $\Phi \lor \Psi=\Phi \cap \Psi$, then $\frak F(\mathcal L)$ is a complete lattice too. Mar 27 revised What is the correspondence between the semantic compactness of logics and the compactness of Stone spaces? added 5 characters in body Mar 27 comment What is the correspondence between the semantic compactness of logics and the compactness of Stone spaces? Let me reconsider it. Mar 27 comment What is the correspondence between the semantic compactness of logics and the compactness of Stone spaces? @MihaHabič Yes. Mar 27 revised What is the correspondence between the semantic compactness of logics and the compactness of Stone spaces? correctify Mar 27 revised What is the correspondence between the semantic compactness of logics and the compactness of Stone spaces? Update Mar 27 revised What is the correspondence between the semantic compactness of logics and the compactness of Stone spaces? Update