# Differences

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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>
+)]