double_stone_algebras - MathStructures

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=====Double Stone algebras=====

Abbreviation: **DblStAlg**
====Definition====
A \emph{double Stone algebra} is a structure $\mathbf{L}=\langle L,\vee ,0,\wedge ,1,^*\rangle $ such that


$\langle L,\vee,0,\wedge,1,^*\rangle $ is a [[Stone algebras]]


$\langle L,\wedge,1,\vee,0,^*\rangle $ is a [[Stone algebras]]

==Morphisms==
Let $\mathbf{L}$ and $\mathbf{M}$ be double Stone algebras. A morphism from $\mathbf{L}$ to $\mathbf{M}$ is a function $h:L\rightarrow M$ that is a
homomorphism: 

$h(x\vee y)=h(x)\vee h(y)$, $h(x\wedge y)=h(x)\wedge h(y)$, $h(0)=0 $, $h(1)=1$, $h(x^*)=h(x)^*$

====Examples====
Example 1: 

====Basic results====

====Properties====
^[[Classtype]]  |variety |
^[[Equational theory]]  | |
^[[Quasiequational theory]]  | |
^[[First-order theory]]  | |
^[[Congruence distributive]]  |yes |
^[[Congruence modular]]  |yes |
^[[Congruence n-permutable]]  | |
^[[Congruence regular]]  | |
^[[Congruence uniform]]  | |
^[[Congruence extension property]]  |yes |
^[[Definable principal congruences]]  | |
^[[Equationally def. pr. cong.]]  | |
^[[Amalgamation property]]  | |
^[[Strong amalgamation property]]  | |
^[[Epimorphisms are surjective]]  | |
^[[Locally finite]]  | |
^[[Residual size]]  | |
====Finite members====

$\begin{array}{lr}
f(1)= &1\\
f(2)= &1\\
f(3)= &1\\
f(4)= &\\
f(5)= &\\
f(6)= &\\
f(7)= &\\
\end{array}$

====Subclasses====
[[Boolean algebras]] 

====Superclasses====
[[Stone algebras]] 


====References====

[(Ln19xx>
)]

Trace: • symmetric_relations • first-order_classes • reflexive_relations • dedekind_categories • continuous_lattices • clifford_semigroups • sequential_algebras • commutative_inverse_semigroups • rings_with_identity • de_morgan_algebras