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 L^{l}_{k}
(L^{l}_{M}) of all closed subsets of
l-valued logic P_{l}, which may be
reflected homomorphically onto P_{k} (onto
subset M⊆P_{k}) 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 P_{k}, was constructed.
2) The set L^{l}_{k}(L^{l}_{M}) contains the noncountable infinite
number of in pairs nonisomorphic classes, if l
≥ k + 2. The set L^{l}_{k}
(L^{l}_{M}) contains the infinite
number of in pairs nonisomorphic classes, if l = k +
1. 3) Let closed subset M⊆P_{k} is
finitely generated, preserves some relation ρ
and contains all constant functions. We proved that
any element of L^{l}_{M} contains
some minimal and is in some maximal element
of L^{l}_{M} 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
L^{l}_{k} and all homomorphisms of
all elements of L^{l}_{k}. 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
⋃^{l}_{k=2}L^{l}_{k} was obtained.