semidistributive_lattices - MathStructures

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=====Semidistributive lattices=====

Abbreviation: **SdLat**
====Definition====
A \emph{semidistributive lattice} is a lattice $\mathbf{L}=\langle L,\vee
,\wedge \rangle $ such that


SD$_{\wedge}$:  $x\wedge y=x\wedge z\Longrightarrow x\wedge y=x\wedge(y\vee z)$


SD$_{\vee}$:  $x\vee y=x\vee z\Longrightarrow x\vee y=x\vee(y\wedge z)$

==Morphisms==
Let $\mathbf{L}$ and $\mathbf{M}$ be semidistributive 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]]  |quasivariety |
^[[Equational theory]]  | |
^[[Quasiequational theory]]  | |
^[[First-order theory]]  |undecidable |
^[[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]]  | |
^[[Locally finite]]  |no |
^[[Residual size]]  |unbounded |
====Finite members====

$\begin{array}{lr}
f(1)= &1\\
f(2)= &1\\
f(3)= &1\\
f(4)= &2\\
f(5)= &4\\
f(6)= &9\\
f(7)= &22\\
f(8)= &60\\
f(9)= &174\\
f(10)= &534\\
f(11)= &1720\\
f(12)= &5767\\
f(13)= &20013\\
f(14)= &71546\\
\end{array}$

====Subclasses====
[[Neardistributive lattices]] 

====Superclasses====
[[Join-semidistributive lattices]] 

[[Meet-semidistributive lattices]] 


====References====

[(Ln19xx>
)]

Trace: • commutative_residuated_partially_ordered_semigroups • commutative_residuated_lattice-ordered_semigroups • commutative_residuated_partially_ordered_monoids • finite_commutative_binars_with_identity_and_zero