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 $s: \mathbb{N} \to L$

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

• 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

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

Properties

Given an element $a \in L$, let $t \mapsto a$ denote the constant sequence to $a$, and given an element $a \in L$ and a sequence $s:\mathbb{N} \to L$, let $T(a, s):\mathbb{N} \to L$ denote the sequence inductively defined by $T(a, s, 0) = a$ and $T(a, s, n + 1) = s(n)$. For simplicity, we denote the infinitary operation as

$$V(s) \coloneqq \Vee_{i \in \mathbb{N}} s(i):(\mathbb{N} \to L) \to L$$

The infinitary operation on a $\sigma$-complete lattice $L$ is

• commutative in that given a sequence $s:\mathbb{N} \to L$ and a bijection $b:\mathbb{N} \cong \mathbb{N}$,

$$V(s) = V(s \circ b)$$

• idempotent in that given an element $a \in L$,

$$V(t \mapsto a) = a$$

• associative in that given an element $a \in L$ and sequences $r:\mathbb{N} \to L$ and $s:\mathbb{N} \to L$,

$$V(T(V(T(a, r)), s)) = V(T(V(T(a, s)), r))$$

i.e. using concatenation to denote the infinitary operation

$$(\ldots r_3 r_2 r_1 r_0 a) s_0 s_1 s_2 s_3 \ldots = \ldots r_3 r_2 r_1 r_0 (a s_0 s_1 s_2 s_3 \ldots)$$

The commutative property being satisfied ensures that all other possible associativity equations are satisfied as well.

• The bottom element $\bot \in L$ is a neutral element of the infinitary operation in the sense that for all elements $a \in L$

$$V(T(a, t \to \bot)) = a$$

The commutative property being satisfied ensures that all other possible neutral element equations are satisfied as well.

See also

• lattice

• sigma-frame

• sigma-complete Heyting algebra

• sigma-complete Boolean algebra

• sigma-continuous valuation

• sigma-continuous probability valuation

• countably prime filter

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)