Franklin's Notes

Lindenbaum algebra

In a first-order language $\mathscr{L}$, we may define an equivalence relation on formulas $\varphi,\psi$ by saying that $\varphi\sim\psi$ iff $\vdash \varphi\leftrightarrow\psi$, and denote the equivalence class of a formula $\varphi$ by writing $(\varphi)$. We may define operations on the equivalence classes as follows:

and we may define constants $0=(\varphi\land\neg\varphi)$ and $1=(\varphi\lor\neg\varphi)$. Finally, if we define $B$ as the set of equivalence classes $(\varphi)$, then the following is a boolean algebra :

and it is called the Lindenbaum algebra of the language $\mathscr{L}$, denoted $\mathfrak{B}_\mathscr{L}$. We may define subalgebras over the sets $B_n$ consisting of formulas with at most the variables $v_1,...,v_n$ free, and denote these subalgebras by $\mathfrak{B}_n$. In particular, $\mathfrak{B}_0$ is the subalgebra of sentences of $\mathscr{L}$.

Suppose $T$ is some theory of a language $\mathscr{L}$, and let $D_T$ be the set of equivalence classes of the elements of $T$, and $D_T'$ be the set of equivalence classes of the consequences of $T$. There is a useful correspondence between semantic properties of $T$ and algebraic properties of $\mathfrak{B}_\mathscr{L}$:

|Theory|Boolean algebra|

|$T$ is consistent|$D_T'$ is a filter|
|$T$ is finitely axiomatizable|$D_T'$ is principal|
|$T$ is complete|$D_T'$ is an ultrafilter|

Question. If $\mathfrak{A}$ is a model of the language $\mathscr{L}$ and $\mathfrak{B}\mathscr{L}$ is a subalgebra of some other boolean algebra $\mathfrak{B}'$, does there necessarily exist a language $\mathscr{L}'$ and a model $\mathfrak{A}'$ such that $\mathfrak{B}'=\mathfrak{B}\mathscr{L}'$ and $\mathfrak{A}'$ is an expansion of $\mathfrak{A}$ to $\mathscr{L}$?




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