shells - MathStructures

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=====Shells=====


====Definition====
A \emph{shell} is a structure $\mathbf{S}=\langle S,+,0,\cdot,1
\rangle $ of type $\langle 2,0,2,0\rangle $ such that


$0$ is an identity for $+$:  $0+x=x$, $x+0=x$


$1$ is an identity for $\cdot$:  $1\cdot x=x$, $x\cdot 1=x$


$0$ is a zero for $\cdot$:  $0\cdot x=0$, $x\cdot 0=0$

==Morphisms==
Let $\mathbf{S}$ and $\mathbf{T}$ be shells. A morphism from $\mathbf{S}$
to $\mathbf{T}$ is a function $h:S\rightarrow T$ that is a homomorphism: 

$h(x+y)=h(x)+h(y)$, $h(x\cdot y)=h(x)\cdot h(y)$, $h(0)=0$, $h(1)=1$

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

====Basic results====

====Properties====
^[[Classtype]]  |variety |
^[[Equational theory]]  |decidable |
^[[Quasiequational theory]]  | |
^[[First-order theory]]  |undecidable |
^[[Locally finite]]  |no |
^[[Residual size]]  |unbounded |
^[[Congruence distributive]]  |no |
^[[Congruence modular]]  |no |
^[[Congruence n-permutable]]  |no |
^[[Congruence regular]]  |no |
^[[Congruence uniform]]  |no |
^[[Congruence extension property]]  |no |
^[[Definable principal congruences]]  |no |
^[[Equationally def. pr. cong.]]  |no |
^[[Amalgamation property]]  |yes |
^[[Strong amalgamation property]]  |yes |
^[[Epimorphisms are surjective]]  | |
====Finite members====

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

====Subclasses====
[[Semirings with identity and zero]] 

====Superclasses====

====References====

[(Ln19xx>
)]

Trace: • de_morgan_algebras • abelian_ordered_groups • goedel_algebras • pure_discriminator_algebras • pseudo_mtl-algebras

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