The expressibility problem in a lattice with closure operation
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N. G. Parvatov
Abstract
In this paper we consider the problem of expressibility of an element of the complete lattice equipped with the closure operation. We find conditions under which this problem has a solution in the form of finite lower neighbourhood. The most interesting case of the complete lattice ℬ(P) of ordered (by inclusion) subsets of the set P is considered separately. It is shown that existence of a finite lower neighbourhood for each generated closed element of this lattice implies finitariness of the operation of closure in it. In the case of the lattice ℬ(P) with finitary closure, we find constructive sufficient conditions for existence of a finite lower neighbourhood of its element. We thus generalise Kuznetsov's theorem on functional completeness. The case of the Galois closure induced by some Galois correspondence between lattices of subsets is studied separately. In this case we find necessary and sufficient conditions for existence of a finite lower neighbourhood of an element of the lattice. This generalises, along with Kuznetsov's theorem, Yablonskii's theorem on predicate-describable classes of functions of finite-valued logic. Along with the expressibility problems, we consider the possibility to determine closed elements of the lattice ℬ(P) in particular, elements of lower neighbourhoods, by finite forbidding sets under some preordering of the set P, and in the case of the Galois closure, by finite descriptions. The theorems we proved are accompanied by examples of their usage.
© de Gruyter 2010
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Articles in the same Issue
- Bounds for the number of Boolean functions admitting affine approximations of a given accuracy
- Identification of a binary Markov chain of order s with r partial connections subjected to additive distortions
- On conditional Internet graphs whose vertex degrees have no mathematical expectation
- An estimate of the probability of localisation of the diameter of a random scale-free graph
- On a recursive class of plateaued Boolean functions
- On the distribution of some characteristics of a quasimonotone mapping
- The expressibility problem in a lattice with closure operation
- An algorithm to restore a linear recurring sequence over the ring R = Zpn from a linear complication of its highest coordinate sequence
- The uniform id-decomposition of functions of many-valued logic over homogeneous functions
- The completeness problem in the function algebra of linear integer-coefficient polynomials
- Circuits for disjunction admitting short unitary diagnostic tests
- The group of automorphisms of the set of bent functions
- Complexity of multiplication in commutative group algebras over fields of prime characteristic
- The summation of Markov sequences on a finite abelian group
- The asymptotics of the number of repetition-free Boolean functions in the basis B1
