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Properties of Boolean functions without three-argument implicents

Abstract Let Fn be the class of all Boolean n-argument functions and let Mn⊂Fn be the class of functions that do not admit implicents of three or less essential variables. Next, let n(3)max (m) be the largest n for which the class Mn contains a function of weight m. It will be shown that n(3)max (11) = 5, n(3)max (12) = 11, n(3)max (13) = 11 (heretofore, the exact values were known only for m ≤ 10). The tensor product of matrices is used to define the binary operation ξ:Fn1 × Fn2 → Fn1-n2. Moreover, it is shown that if one of the arguments ξ belongs to Mn1 and the other is a Boolean function f (x1, ..., xn2) such that f(x1, ...,xn2)⋁f(x̄1, ...,x̄n2) Belongs to Mn2, then the result of the operation always belongs to Mn1-n2. With relaxed conditions, the result of ξ may fail to belong to Mn1-n2. The operation ξ is used in the algorithm for construction of a Boolean functions without three-argument implicents with arbitrary small ratio of the weight of the function to the number of its variables.