Abstract
The main goal of this chapter is to introduce several basic concepts in algebraic logic, i.e., Lindenbaum-Tarski algebras, locally finite algebras, finite embeddability property and canonical extensions. They are important algebraic tools for developing algebraic approach to logic.
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Notes
- 1.
The notion was originally introduced by Heyting (1930).
- 2.
The algebra was named after A. Lindenbaum and A. Tarski. It was introduced firstly in Tarski (1935, 1936).
- 3.
We exclude “empty” algebras. For Heyting algebras, the subalgebra \(\mathbf C\) generated by any subset X of A is always non-empty, since 0 must be an element of C. But, the same statement does not necessarily hold, as some algebras may not have constants in the definition. In such a case, we need to assume that a set X of generators is non-empty. Such an example of a class of algebras is treated in Remark 7.4 below.
- 4.
As we mentioned in Example 6.7, the word sublattices is used instead of subalgebras, as we are discussing lattices.
- 5.
The fact was discovered independently by Rieger (1949) and Nishimura (1960).
- 6.
See also the footnote 1 in Sect. 5.3.
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Ono, H. (2019). Basics of Algebraic Logic. In: Proof Theory and Algebra in Logic. Short Textbooks in Logic. Springer, Singapore. https://doi.org/10.1007/978-981-13-7997-0_7
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DOI: https://doi.org/10.1007/978-981-13-7997-0_7
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