almost_distributive_lattices

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=====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>
)]

Trace: • intuitionistic_linear_logic_algebras • distributive_p-algebras • flc-algebras • ordered_semigroups • algebraic_lattices • frames • representable_lattice-ordered_groups • stone_algebras • multiplicative_semilattices • nonassociative_relation_algebras