Lattices & Permutation Groups (algebrax.lattice & algebrax.group)¶

AlgebraX provides discrete algebraic structures:

Lattice Operations (algebrax.lattice)¶

• join(a, b): Least Upper Bound (LUB) supremum \(a \vee b\).
• meet(a, b): Greatest Lower Bound (GLB) infimum \(a \wedge b\).
• combine(a, b, op): Element-wise vector and coordinate map reduction.

Permutation Groups (\(S_n\)) (algebrax.group)¶

• compose(p1, p2): Function composition of permutations \((\pi_2 \circ \pi_1)(x) = \pi_2(\pi_1(x))\).
• signature(p): Permutation parity signature \(\operatorname{sgn}(\pi) \in \{+1, -1\}\).
• inverse(p): Group inverse permutation \(\pi^{-1}\).

Related Recipes & Applications¶

• Distributed Vector Clocks — Join-semilattice synchronization.
• Algebraic Knot Theory — Artin braid permutations and parity signatures via group.compose and group.signature.
• Vibration Structural Analysis — Rotor permutation symmetries via group.compose.