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complemented_lattices [2010/07/29 15:46] (current)
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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>
 +)]