This article is cited in 4 papers

On implicit extensions in many-valued logic

S. S. Marchenkov

Lomonosov Moscow State University

Abstract: We consider Kuznetsov's implicit expressibility and its generalizations, when the implicit expressibility language is augmented with the additional disjunction, implication, and negation logical connectives. It is shown that, for each $k\geqslant 3$, the implicit extensions in $P_k$ have the cardinality of the continuum. For each $k\geqslant 3$, we also prove that each of the sets of positively implicit, implicatively implicit, and negatively implicit extensions in $P_k$ contains, respectively, as a proper subset, the set of positively implicit, implicatively implicit, and negatively implicit closed classes. We verify that, for $k\geqslant 2$, the functions of the set $H_k^*$ of homogeneous functions preserving the set $E_{k-1}$ can be used for producing implicatively implicit and negatively implicit extensions without changing the result.

Keywords: implicit extension, many-valued logic.

UDC: 519.716

Received: 23.01.2023

DOI: 10.4213/dm1764

 English version: Discrete Mathematics and Applications, 2024, 34:5, 277–282

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