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On unit-zero Boolean representations of operations and relations. (English) Zbl 0006.00404

In an algebra \((K,+,\times)\), such as ordinary real algebra, in which there are two elements \(0\) and \(1\) with the usual properties \(a+0=0+a=a\) and \(a 1 = 1 a = a\), let \((x_1, x_2, \dots, x_m; a_1, a_2, \dots, a_m)\) denote a unit-zero function with respect to the sequence of \(m\) elements \(a_1, a_2, \dots, a_m\) of \(K\), that is, a function \(f(x_1, x_2, \dots, x_m)\) which has the value \(1\) or \(0\) according as the equalities \(x_i=a_i\) \(i=1, 2, \dots, m\), all hold or do not all hold. If \(f = 0\) or \(1\) in the respective cases then \(f\) is said to be a zero-unit function. An algebra which has a unit-zero [zero-unit] function for every sequence of \(m\) of its elements is called a unit-zero [zero-unit] algebra. The principal results of the paper are the following two dual propositions. The only Boolean unit-zero algebra is a two-element Boolean algebra. The only zero-unit Boolean algebra is a two-element Boolean algebra.

MSC:

06E05 Structure theory of Boolean algebras
08A02 Relational systems, laws of composition
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