Geodesic
An algebraic completeness proof for Kleene's 3-valued logic
Bollettino della Unione matematica italiana, Série 8, 5B (2002) no. 2, pp. 447-467

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We introduce Kleene's 3-valued logic in a language containing, besides the Boolean connectives, a constant $n$ for the undefined truth value, so in developing semantics we can switch from the usual treatment based on DM-algebras to the narrower class of DMF-algebras (De Morgan algebras with a single fixed point for negation). A sequent calculus for Kleene's logic is introduced and proved complete with respect to threevalent semantics. The completeness proof is based on a version of the prime ideal theorem that is typical of DMF-algebras. Only for the weak completeness theorem the proof is fully algebrical, because in the proof of strong completeness we have been compelled to use topological methods (Tychonoff theorem on the product of compact spaces). 
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     author = {Negri, Maurizio},
     title = {An algebraic completeness proof for {Kleene's} 3-valued logic},
     journal = {Bollettino della Unione matematica italiana},
     pages = {447--467},
     publisher = {mathdoc},
     volume = {Ser. 8, 5B},
     number = {2},
     year = {2002},
     zbl = {1096.03021},
     mrnumber = {MR1911200},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/BUMI_2002_8_5B_2_a9/}
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Negri, Maurizio. An algebraic completeness proof for Kleene's 3-valued logic. Bollettino della Unione matematica italiana, Série 8, 5B (2002) no. 2, pp. 447-467. http://geodesic.mathdoc.fr/item/BUMI_2002_8_5B_2_a9/