# Representation of Boolean algebras

Stone's representation theorem states that every Boolean algebra is isomorphic to an algebra of point sets.

Loomis-Sikorski theorem states that ''every $\sigma$-complete Boolean algebra is $\sigma$-isomorphic to a $\sigma$-complete Boolean algebra of point sets modulo a $\sigma$-ideal in that algebra (Loomis, 1947)''.

Why do we have to take the quotient space of a $\sigma$-complete Boolean algebra of point sets by a $\sigma$-ideal in that algebra for the representation of $\sigma$-complete Boolean algebras?

Thanks a lot for clarifying.

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