complemented_lattices - MathStructures

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=====Complemented lattices=====

Abbreviation: **CdLat**
====Definition====
A \emph{complemented lattice} is a [[bounded lattices]] $\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 complemented lattices. A morphism from $\mathbf{L}$ to $\mathbf{M}$ is a function $h:L\rightarrow M$ 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 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]]  | |
^[[First-order theory]]  |undecidable |
^[[Locally finite]]  |no |
^[[Residual size]]  |unbounded |
^[[Congruence distributive]]  |yes |
^[[Congruence modular]]  |yes |
^[[Congruence n-permutable]]  |yes |
^[[Congruence regular]]  |no |
^[[Congruence uniform]]  |no |
^[[Congruence extension property]]  |no |
^[[Definable principal congruences]]  |no |
^[[Equationally def. pr. cong.]]  |no |
^[[Amalgamation property]]  | |
^[[Strong amalgamation property]]  | |
^[[Epimorphisms are surjective]]  | |
====Finite members====

$\begin{array}{lr}
f(1)= &1\\
f(2)= &1\\
f(3)= &0\\
f(4)= &1\\
f(5)= &2\\
f(6)= &\\
f(7)= &\\
f(8)= &\\
\end{array}$

====Subclasses====
[[Complemented modular lattices]] 

====Superclasses====
[[Bounded lattices]] 


====References====

[(Ln19xx>
)]

Trace: • abelian_lattice-ordered_groups • t0-spaces • right_hoops • distributive_allegories • finite_commutative_semigroups • distributive_lattices • complemented_lattices • division_rings • ore_domains • ordered_semilattices