stone_algebras - MathStructures

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=====Stone algebras=====

Abbreviation: **StAlg**
====Definition====
A \emph{Stone algebra} is a [[distributive p-algebra]] $\mathbf{L}=\langle L,\vee ,0,\wedge ,1,^*\rangle $ such that

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

==Morphisms==
Let $\mathbf{L}$ and $\mathbf{M}$ be 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====
^[[Equational theory]]  |decidable |
^[[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]]  |yes |
^[[Strong amalgamation property]]  | |
^[[Epimorphisms are surjective]]  | |
^[[Locally finite]]  | |
^[[Residual size]]  | |
====Finite members====

$\begin{array}{lr}
f(1)= &1\\
f(2)= &1\\
f(3)= &1\\
f(4)= &2\\
f(5)= &2\\
f(6)= &4\\
f(7)= &5\\
f(8)= &10\\
f(9)= &16\\
f(10)= &28\\
\end{array}$

====Subclasses====
[[Double Stone algebras]] 

====Superclasses====
[[Distributive p-algebras]] 


====References====

[(Ln19xx>
)]

Trace: • stone_algebras • lie_algebras • bck-algebras • modal_algebras • commutative_idempotent_integral_involutive_fl-algebras • quandles • join-semidistributive_lattices • ordered_semilattices • distributive_allegories • semirings_with_identity