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almost_distributive_lattices [2010/07/29 15:12] jipsen created |
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- | f | + | =====Almost distributive lattices===== |

+ | | ||

+ | Abbreviation: **ADLat** | ||

+ | | ||

+ | ====Definition==== | ||

+ | An \emph{almost distributive lattice} is a [[neardistributive lattice]] $\mathbf{L}=\langle L,\vee,\wedge\rangle$ such that | ||

+ | | ||

+ | AD$_{\wedge}$: $v\wedge[u\vee (x\wedge[y\vee (x\wedge z)])]\le u\vee [(x\wedge[y\vee (x\wedge z)])\wedge(v\vee (x\wedge y)\vee (x\wedge z))]$ | ||

+ | | ||

+ | AD$_{\vee}$: $v\vee[u\wedge (x\vee[y\wedge (x\vee z)])]\ge u\wedge [(x\vee[y\wedge (x\vee z)])\vee(v\wedge (x\vee y)\wedge (x\vee z))]$ | ||

+ | | ||

+ | ==Morphisms== | ||

+ | Let $\mathbf{L}$ and $\mathbf{M}$ be almost distributive lattices. A morphism from $\mathbf{L}$ to $\mathbf{M}$ is a function | ||

+ | $h:L\rightarrow M$ that is a homomorphism: | ||

+ | | ||

+ | $h(x\vee y)=h(x)\vee h(y)$, $h(x\wedge y)=h(x)\wedge h(y)$ | ||

+ | | ||

+ | ====Examples==== | ||

+ | Example 1: $D[d]=\langle D\cup\{d'\},\vee ,\wedge\rangle$, where $D$ is any distributive lattice and $d$ is an element in it that | ||

+ | is split into two elements $d,d'$ using Alan Day's doubling construction. | ||

+ | | ||

+ | | ||

+ | ====Basic results==== | ||

+ | | ||

+ | | ||

+ | ====Properties==== | ||

+ | ^[[Classtype]] |variety | | ||

+ | ^[[Equational theory]] | | | ||

+ | ^[[Quasiequational theory]] | | | ||

+ | ^[[First-order theory]] |undecidable | | ||

+ | ^[[Locally finite]] |no | | ||

+ | ^[[Residual size]] |unbounded | | ||

+ | ^[[Congruence distributive]] |yes | | ||

+ | ^[[Congruence modular]] |yes | | ||

+ | ^[[Congruence n-permutable]] |no | | ||

+ | ^[[Congruence regular]] |no | | ||

+ | ^[[Congruence uniform]] |no | | ||

+ | ^[[Congruence extension property]] | | | ||

+ | ^[[Definable principal congruences]] | | | ||

+ | ^[[Equationally def. pr. cong.]] | | | ||

+ | ^[[Amalgamation property]] |no | | ||

+ | ^[[Strong amalgamation property]] |no | | ||

+ | ^[[Epimorphisms are surjective]] | | | ||

+ | | ||

+ | ====Finite members==== | ||

+ | | ||

+ | $\begin{array}{lr} | ||

+ | f(1)= &1\\ | ||

+ | f(2)= &1\\ | ||

+ | f(3)= &1\\ | ||

+ | f(4)= &2\\ | ||

+ | f(5)= &4\\ | ||

+ | f(6)= &\\ | ||

+ | f(7)= &\\ | ||

+ | \end{array}$ | ||

+ | | ||

+ | | ||

+ | ====Subclasses==== | ||

+ | [[Distributive lattices]] | ||

+ | | ||

+ | | ||

+ | ====Superclasses==== | ||

+ | [[Neardistributive lattices]] | ||

+ | | ||

+ | | ||

+ | ====References==== | ||

+ | | ||

+ | [(Ln19xx> | ||

+ | )] |

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