nLab sigma-complete lattice

Definition

A $\sigma$-complete lattice is a lattice $(L, \leq, \bot, \vee, \top, \wedge)$ with a function

$$\Vee_{i:\mathbb{N}} (-)(i): (\mathbb{N} \to L) \to L$$

such that

• for all natural numbers $n \in \mathbb{N}$ and sequences $s: \mathbb{N} \to L$

  $$s(n) \leq \Vee_{i:\mathbb{N}} s(i)$$

• for all elements $x \in L$ and sequences $s: \mathbb{N} \to L$, if $s(n) \leq x$ for all natural numbers $n \in \mathbb{N}$, then

  $$\Vee_{i:\mathbb{N}} s(i) \leq x$$

See also

• lattice

• sigma-frame

• sigma-continuous valuation

• sigma-continuous probability valuation

References

• Alex Simpson, Measure, randomness and sublocales, Annals of Pure and Applied Logic, Volume 163, Issue 11, November 2012, Pages 1642-1659. (doi:10.1016/j.apal.2011.12.014)