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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> | ||

+ | )] | ||

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