Lattice

A lattice is a partially ordered set in which every pair has a greatest lower bound, called its meet, and a least upper bound, called its join.

In a lattice, joins and meets are the pairwise least upper bounds and greatest lower bounds.

The join of two elements of a lattice is their least upper bound. In a lattice of closed set families, it is the closure of their union.

The meet of two elements of a lattice is their greatest lower bound. For set families ordered by inclusion, it is their intersection whenever that intersection lies in the specified lattice.

A homomorphism between lattices preserves binary joins and meets: $f(x\vee y)=f(x)\vee f(y)$ and $f(x\wedge y)=f(x)\wedge f(y)$. It is order preserving, since $x\le y$ implies $x=x\wedge y$ and hence $f(x)=f(x)\wedge f(y)$. For a map into the two-element lattice, the inverse image of zero is a lattice ideal, and meet preservation says this ideal is prime: if a meet belongs to it, at least one factor does. Conversely the indicator of the complement of a prime lattice ideal preserves both operations. Preservation of distinguished top and bottom elements is an additional requirement for homomorphisms of bounded lattices.

A complete lattice is a partially ordered set in which every subset has a meet and a join, including the empty subset.

Every order-preserving function from a complete lattice to itself has a least and a greatest fixed point. More strongly, its set of fixed points is itself a complete lattice.

A distributive lattice is a lattice in which meet distributes over join and, equivalently, join distributes over meet.

In a distributive lattice,

A Boolean algebra is a bounded distributive lattice in which every element has a complement. Its meet, join, and complement abstract conjunction, disjunction, and negation.

Every proper lattice ideal of a Boolean algebra extends to a prime lattice ideal. More generally, an ideal disjoint from a lattice filter extends to a prime ideal still disjoint from that filter. This Boolean case of the Stone prime filter theorem is equivalent to the ultrafilter lemma over Zermelo–Fraenkel set theory and is weaker than the axiom of choice.

The lattice ideals of a Boolean algebra form a frame under inclusion. Meets are intersections and joins are ideals generated by unions. A complemented ideal is principal: if $I\vee J=B$ and $I\cap J=\{0\}$, choose $b\in I,c\in J$ with $b\vee c=1$; then $c=\neg b$ and $I=\downarrow b$. This ideal frame defines an extremally disconnected locale exactly when $B$ is complete.

A Boolean algebra in which every set-indexed family has a join and a meet. Double-negation Boolean algebra is an intuitionistic example, with joins formed by regularizing unions. Completeness makes truth values for quantifiers possible in a Boolean-valued inner universe over intuitionistic set theory.

Inside IZF, truth values are subsets of a singleton. Their double-negation-stable elements form a complete Boolean algebra: arbitrary meets are intersections and arbitrary joins are double-negation closures of unions. Negation gives Boolean complement. The join of $p$ and its complement is top although their ordinary union need not be top. The closure here is taken inside the fixed singleton; it does not assume that the class of all objects double-negatively belonging to an arbitrary set is itself a set.

A lattice filter is a nonempty upward-closed subset closed under finite meets. A proper filter omits the bottom element.

A lattice filter is a filter in an ordered set when that order is a lattice; directedness and upward closure then imply closure under finite meets.

A prime filter is a proper lattice filter $P$ such that $a\vee b\in P$ implies $a\in P$ or $b\in P$.

If a lattice filter $F$ and lattice ideal $I$ in a distributive lattice are disjoint, there is a prime filter of a distributive lattice containing $F$ and disjoint from $I$.

The Boolean prime ideal theorem is the Boolean-algebra version; distributivity gives this corresponding filter-extension statement.

A lattice ideal is a nonempty downward-closed subset closed under finite joins. It is the order-dual notion to a lattice filter.

A proper lattice ideal $I$ is prime if $x\wedge y\in I$ implies $x\in I$ or $y\in I$. Its complement is a prime filter of a distributive lattice when the lattice is distributive: downward closure of the ideal makes the complement upward closed, primeness makes it meet-closed, and join closure makes that filter prime. Equivalently, its complement indicator is a homomorphism of lattices to the two-element lattice. Given $b\nleqq a$ in a distributive lattice, maximize an ideal containing $a$ and avoiding $b$ using Zorn lemma. If $x,y$ lie outside it, the two enlarged ideals contain $b$, so $b\le i\vee x$ and $b\le j\vee y$ for some $i,j\in I$. Distributivity shows that $x\wedge y\in I$ would force $b\in I$, a contradiction. Thus this maximal avoiding ideal is prime and separates the two specified elements.

Priestley duality is a dual equivalence between bounded distributive lattices and compact totally order-disconnected ordered topological spaces.

The Priestley dual space $\hat{L}$ of a bounded distributive lattice $L$ consists of its prime filters, ordered by inclusion and topologized by the sets $a^{\ast }=\{P:a\in P\}$ and their complements.

This construction yields a Priestley space: a compact ordered topological space in which suitable clopen upward-closed sets separate distinct comparable-order failures.

A clopen up-set is both clopen in the topology and upward closed in the order. The clopen up-sets of a Priestley space form a distributive lattice under intersection and union.

The Stone map sends an element $a$ of a distributive lattice to the clopen up-set $a^{\ast }=\{P:a\in P\}$ of its Priestley dual space. The Stone prime filter theorem makes this lattice homomorphism injective.