Definition:Bounded Lattice/Definition 2

From ProofWiki
Jump to: navigation, search

Definition

Let $\left({S, \vee, \wedge, \preceq}\right)$ be a lattice.

Let $\vee$ and $\wedge$ have identity elements $\bot$ and $\top$ respectively.


Then $\left({S, \vee, \wedge, \preceq}\right)$ is a bounded lattice.


Thus $\left({S, \vee, \wedge}\right)$ is a bounded lattice iff the following axioms are satisfied:

\((L0)\)   $:$   Closure      \(\displaystyle \forall a, b:\) \(\displaystyle a \vee b \in S \)    \(\displaystyle a \wedge b \in S \)             
\((L1)\)   $:$   Commutativity      \(\displaystyle \forall a, b:\) \(\displaystyle a \vee b = b \vee a \)    \(\displaystyle a \wedge b = b \wedge a \)             
\((L2)\)   $:$   Associativity      \(\displaystyle \forall a, b, c:\) \(\displaystyle a \vee \left({b \vee c}\right) = \left({a \vee b}\right) \vee c \)    \(\displaystyle a \wedge \left({b \wedge c}\right) = \left({a \wedge b}\right) \wedge c \)             
\((L3)\)   $:$   Idempotence      \(\displaystyle \forall a:\) \(\displaystyle a \vee a = a \)    \(\displaystyle a \wedge a = a \)             
\((L4)\)   $:$   Absorption      \(\displaystyle \forall a,b:\) \(\displaystyle a \vee \left({a \wedge b}\right) = a \)    \(\displaystyle a \wedge \left({a \vee b}\right) = a \)             
\((L5)\)   $:$   Identity elements      \(\displaystyle \exists \top, \bot: \forall a:\) \(\displaystyle a \vee \bot = a = \bot \vee a \)    \(\displaystyle a \wedge \top = a = \top \vee a \)