Blattice

Blattice is a term used in several disciplines to describe a lattice-like structure that carries an additional boolean annotation on its elements. There is no single canonical definition; the exact meaning depends on context and author.

In formal lattice theory, a common formalization treats Blattice as a lattice L equipped with a labeling function b: L -> {0,1} that is compatible with the lattice operations, for example a lattice homomorphism where b(x ∨ y) = b(x) ∨ b(y) and b(x ∧ y) = b(x) ∧ b(y). The labeling thus encodes a two-valued state on elements and can interact with order relations. Such a structure is used to reason about presence/absence, enable pruning in algorithms, or represent truth-assignments in logical frameworks.

In practical applications, Blattice appears in data structures and computer graphics to model multi-resolution or hierarchical grids where each cell carries an on/off state. It also appears in constraint solving and formal concept analysis as a way to combine lattice operations with boolean features. In some descriptions, a Blattice may refer to a boolean lattice arranged with additional dimensions, or to a lattice augmented by an auxiliary boolean layer to enable efficient evaluation or indexing.

A simple example is the power set lattice P(S) with the subset inclusion order, together with the labeling b(A) = 1 if A is nonempty and b(A) = 0 otherwise. This labeling preserves join and meet and yields a bona fide two-valued lattice homomorphism.

The term is largely descriptive and not tied to a single standard definition; readers encountering Blattice should consult the source for the precise axioms and intended use in that text.