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=2L^l _k was obtained.

many-valued logic, closed class, homomorphism, limitative logic

Mathematical problems and theory of numerical methods