The Lindenbaum algebra is a natural Boolean algebra associated with any theory $T$. The Lindenbaum algebra can be taken to consist of equivalence classes of formulas, where two formula are equivalent if they are proved equivalent by $T$, and the Boolean algebra structure is inherited naturally from the syntax. Since the Lindenbaum algebra is a Boolean algebra, it admits of diverse characterizations in mathematics, for every Boolean algebra can be viewed altnertively alternatively as a ring (a Boolean ring), as an algebraic structure (with $\wedge$ and $\vee$), as a partial order (defined by an order $\leq$ with certain properties)properties, such as lub, glb and complements), as a lattice (either with $\wedge$ and $\vee$ or $\leq$) or finally, as a category (since every partial order can be viewed as a category).
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The Lindenbaum algebra is a natural Boolean algebra associated with any theory $T$. The Lindenbaum algebra can be taken to consist of equivalence classes of formulas, where two formula are equivalent if they are proved equivalent by $T$, and the Boolean algebra structure is inherited naturally from the syntax. Since the Lindenbaum algebra is a Boolean algebra, it admits of diverse characterizations in mathematics, for every Boolean algebra can be viewed altnertively as a ring (a Boolean ring), as an algebraic structure (with $\wedge$ and $\vee$), as a partial order (defined by $\leq$ with certain properties), as a lattice (either with $\wedge$ and $\vee$ or $\leq$) or finally, as a category (since every partial order can be viewed as a category). |
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