Advanced Łukasiewicz calculus and MV-algebras by D. Mundici

Author:D. Mundici
Language: eng
Format: epub
Publisher: Springer Netherlands, Dordrecht

11.1 Basically Disconnected Spaces

An MV-algebra is - if every countable set of elements of has a least upper bound (also known as a sup, or supremum) with respect to the underlying lattice order of Thus for each and whenever is an upper bound of all the it follows that We write to mean that is the sup of the in The ambient MV-algebra will always be clear from the context.

By (A21.38) every -complete MV-algebra is semisimple.

Given two -complete MV-algebras and a homomorphism is called a - if for each sequence the image of the sup in of the coincides with the sup in of the sequence

Every finite MV-algebra is -complete—indeed, is in the sense that every nonempty subset of has a supremum in The standard example of a complete MV-algebra is Further examples of -complete MV-algebras are given by -complete boolean algebras.

A trivial adaptation of the proof of (A21.39) in [1, p. 130] shows that every -complete MV-algebra satisfies the countable distributivity law

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