By David C. Rine
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Additional info for Computer Science and Multiple-Valued Logic. Theory and Applications
However the next theorem shows that this infinite distributivity law holds in any Post algebra L. Theorem 24. The infinite distributive law y VIG/JC, = V,eIyjc, is valid in any Post algebra L whenever x = V,e/jc, exists. The dual statement is also true. Proof. ), B,iel B;iel since the law is valid in any Boolean algebra. Hence, by Theorem 11, Dj(yx) = VB;«e/ Dj(yXi) for each / = 1 , . . , n - 1 , and yjc = VY;,e/yjc, by Theorem 22. The duality theorem yields the dual result. Theorem 25. If x = V,ej JC, exists, then the following generalized rules of simplifi cation, stated in recursive form, are valid: (1) Cn_,(jc)= V C,-,(*), Ch.
A ring theoretic function f(x, y) in R is determined by the p2 values of fii, j), where i = 0, 1 , . . , p - 1 and j = 0, 1 , . . , p - 1. Let Mix, y) and mix, y) be those ring theoretic functions for which Mii, j) = max(/, /) and m (/, /) = min(i, /), where i = 0, 1 , . . , p - 1 and j = 0, 1 , . . , p - 1. Thus, for example, in the case p = 3, M(jc,y) = jc + y +jc2-y 4-2-Jc-y + jc2-y2 + xy2, mix, y) = 2-x2-y +x-y +2-JC2-y2 + 2-JC-y2, where the operations involved are the ring operations of addition and multiplica tion.
That is, if the Post algebra L is complete, then L is a topological lattice[l, p. 146]. However the next theorem shows that this infinite distributivity law holds in any Post algebra L. Theorem 24. The infinite distributive law y VIG/JC, = V,eIyjc, is valid in any Post algebra L whenever x = V,e/jc, exists. The dual statement is also true. Proof. ), B,iel B;iel since the law is valid in any Boolean algebra. Hence, by Theorem 11, Dj(yx) = VB;«e/ Dj(yXi) for each / = 1 , . . , n - 1 , and yjc = VY;,e/yjc, by Theorem 22.