Bernstein, B. A. On unit-zero Boolean representations of operations and relations. (English) Zbl 0006.00404 Bull. Am. Math. Soc. 38, 707-712 (1932). 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. Reviewer: R. D. Carmichael (Urbana) Page: −3 −2 −1 ±0 +1 +2 +3 +4 +5 Show Scanned Page MSC: 06E05 Structure theory of Boolean algebras 08A02 Relational systems, laws of composition Keywords:zero-unit function; zero-unit Boolean algebra; two-element Boolean algebra PDFBibTeX XMLCite \textit{B. A. Bernstein}, Bull. Am. Math. Soc. 38, 707--712 (1932; Zbl 0006.00404) Full Text: DOI