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Article collection "Mathematical Problems of Cybernetics" №4, Moscow, 1992
Authors: Makarov A.V.
On homomorphisms of functional systems of many-valued logics.
Abstract:
The properties of the set Llk (LlM) of all closed subsets of l-valued logic Pl, which may be reflected homomorphically onto Pk (onto subset M⊆Pk) are investigated. The main results. 1) We corrected and improved the well known Demetrovich's results. The following properties of limitative logics were proved: the existence of the limitative logic with the empty table, the nontransitivity of the relation of absorbtion for limitative logic. Also the noncountable infinite set of in pairs individing each other limitative logic was constructed. Also the hypercontinual infinite set of in pairs individing each other closed subsets of countable - valued logics, which may be reflected homomorphically onto Pk, was constructed. 2) The set Llk(Ll M) contains the noncountable infinite number of in pairs nonisomorphic classes, if l ≥ k + 2. The set Llk (LlM) contains the infinite number of in pairs nonisomorphic classes, if l = k + 1. 3) Let closed subset M⊆Pk is finitely generated, preserves some relation ρ and contains all constant functions. We proved that any element of LlM contains some minimal and is in some maximal element of LlM contained. The set of all maximal elements is the finite set and any maximal element has finite number of homomorphisms onto M. Any minimal element is finitely generated. 4) We determined all maximal elements of Llk and all homomorphisms of all elements of Llk. We proved that any maximal element is generated by the only function. The asymptotic formula for the number of in pairs nonisomorphic maximal elements of ⋃lk=2Ll k was obtained.
Keywords:
many-valued logic, closed class, homomorphism, limitative logic
Publication language: russian,  pages: 25 (p. 5-29)
Research direction:
Mathematical problems and theory of numerical methods
Russian source text:
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