Boolean-valued algebras

V. N. Salii

Sbornik. Mathematics, Tome 21 (1973) no. 4, pp. 544-557

Résumé

The paper contains the construction of a general theory of Boolean-valued algebras: There are introduced the notions of a homeomorphism, congruence, subalgebra and direct product. It is shown that these algebras possess properties that are totally analogous to the properties of two-valued algebras. To every Boolean-valued algebra $\mathfrak A$ there is related a certain universal algebra $\mathfrak{N(A)}$, called the normal extension of $\mathfrak A$, whose elements are all the partitions of unity of the given Boolean algebra, with naturally extended operations. The equational equivalence of an arbitrary Boolean-valued algebra and its normal extension is proved. It is shown that every homomorphism of a Boolean-valued algebra can be uniquely extended to a homomorphism of its normal extension. Bibliography: 10 titles.