boolean_lattices - MathStructures

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=====Boolean lattices=====

Abbreviation: **BoolLat**
====Definition====
A \emph{Boolean lattice} is a 
[[bounded distributive lattice]] 
$\mathbf{L}=\langle L,\vee ,0,\wedge ,1\rangle $ such that

every element has a complement:  $\exists y(x\vee y=1\mbox{ and }x\wedge y=0)$

==Morphisms==
Let $\mathbf{L}$ and $\mathbf{M}$ be bounded distributive lattices. 
A morphism from $\mathbf{L}$ to $\mathbf{M}$ is a function $h:L\to M$ that is a
bounded lattice homomorphism: 

$h(x\vee y)=h(x)\vee h(y)$, $h(x\wedge y)=h(x)\wedge h(y)$, $h(0)=0 $, $h(1)=1$

====Examples====
Example 1: $\langle \mathcal P(S), \cup, \emptyset, \cap, S\rangle$, the collection
of subsets of a set $S$, with union, empty set, intersection, and the whole
set $S$.


====Basic results====


====Properties====
^[[Classtype]]  |first-order |
^[[Equational theory]]  |decidable |
^[[Quasiequational theory]]  |decidable |
^[[First-order theory]]  |decidable |
^[[Congruence distributive]]  |yes |
^[[Congruence modular]]  |yes |
^[[Congruence n-permutable]]  |yes |
^[[Congruence regular]]  |yes |
^[[Congruence uniform]]  |yes |
^[[Congruence extension property]]  |yes |
^[[Definable principal congruences]]  |yes |
^[[Equationally def. pr. cong.]]  | |
^[[Amalgamation property]]  | |
^[[Strong amalgamation property]]  | |
^[[Epimorphisms are surjective]]  | |
^[[Locally finite]]  |yes |
^[[Residual size]]  | |

====Finite members====
Any finite member is a power of the 2-element Boolean lattice.


====Subclasses====
[[Boolean algebras]]

====Superclasses====
[[Complemented modular lattices]] 

[[Bounded distributive lattices]] 


====References====

[(Ln19xx>
)]

Trace: • small_categories • categories • commutative_binars • finite_monoids_with_zero • basic_logic_algebras • first-order_formulas • finite_binars_with_zero • distributive_lattice_ordered_semigroups • congruence_uniform • nonassociative_algebra