Properties of quasi-Boolean function on quasi-Boolean algebra

Yang-jin Cheng () and Lin-xi Xu
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Yang-jin Cheng: Xiangtan University
Lin-xi Xu: Xiangtan University

Fuzzy Information and Engineering, 2011, vol. 3, issue 3, 275-291

Abstract: Abstract In this paper, we investigate the following problem: give a quasi-Boolean function Ψ(x 1, …, x n ) = (a ∧ C) ∨ (a 1 ∧ C 1) ∨ … ∨ (a p ∧ C p ), the term (a ∧ C) can be deleted from Ψ(x 1, …, x n )? i.e., (a ∧ C) ∨ (a 1 ∧ C 1) ∨ … ∨ (a p ∧ C p ) = (a 1 ∧ C 1) ∨ … ∨ (a p ∧ C p )? When a = 1: we divide our discussion into two cases. (1) ℑ1(Ψ,C) = ø, C can not be deleted; ℑ1(Ψ,C) ≠ ø, if S i 0 ≠ ø (1 ≤ i ≤ q), then C can not be deleted, otherwise C can be deleted. When a = m: we prove the following results: (m∧C)∨(a 1∧C 1)∨…∨(a p ∧C p ) = (a 1∧C 1)∨…∨(a p ∧C p ) ⇔ (m ∧ C) ∨ C 1 ∨ … ∨C p = C 1 ∨ … ∨C p . Two possible cases are listed as follows, (1) ℑ2(Ψ,C) = ø, the term (m∧C) can not be deleted; (2) ℑ2(Ψ,C) ≠ ø, if (∃i 0) such that $S'_{i_0 } $ = ø, then (m∧C) can be deleted, otherwise ((m∧C)∨C 1∨…∨C q )(v 1, …, v n ) = (C 1 ∨ … ∨ C q )(v 1, …, v n )(∀(v 1, …, v n ) ∈ L 3 n ) ⇔ (C 1 ′ ∨ … ∨ C q ′ )(u 1, …, u q ) = 1(∀(u 1, …, u q ) ∈ B 2 n ).

Keywords: Lattice; Boolean function; Quasi-Boolean algebra; Quasi-Boolean function (search for similar items in EconPapers)
Date: 2011
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DOI: 10.1007/s12543-011-0083-8

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