Axiom:Lattice Axioms
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Definition
Let $\struct {S, \vee, \wedge, \preceq}$ be an ordered structure.
$\struct {S, \vee, \wedge, \preceq}$ is a lattice if and only if the following axioms are satisfied:
\((\text L 0)\) | $:$ | Closure | \(\ds \forall a, b:\) | \(\ds a \vee b \in S \) | \(\ds a \wedge b \in S \) | ||||
\((\text L 1)\) | $:$ | Commutativity | \(\ds \forall a, b:\) | \(\ds a \vee b = b \vee a \) | \(\ds a \wedge b = b \wedge a \) | ||||
\((\text L 2)\) | $:$ | Associativity | \(\ds \forall a, b, c:\) | \(\ds a \vee \paren {b \vee c} = \paren {a \vee b} \vee c \) | \(\ds a \wedge \paren {b \wedge c} = \paren {a \wedge b} \wedge c \) | ||||
\((\text L 3)\) | $:$ | Idempotence | \(\ds \forall a:\) | \(\ds a \vee a = a \) | \(\ds a \wedge a = a \) | ||||
\((\text L 4)\) | $:$ | Absorption | \(\ds \forall a,b:\) | \(\ds a \vee \paren {a \wedge b} = a \) | \(\ds a \wedge \paren {a \vee b} = a \) |
These criteria are called the lattice axioms.